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import Mathlib.Data.Real.EReal
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Bochner

import NMDSProofs.Utils
import NMDSProofs.RKHS
import NMDSProofs.PushForward
import NMDSProofs.SteepestDirection

local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y)

open scoped RealInnerProductSpace
open BigOperators Finset ENNReal NNReal MeasureTheory

set_option trace.Meta.Tactic.simp.rewrite true
set_option maxHeartbeats 400000

/-
  We defined measures μ and π (ν is considered as the standard Lebesgue measure) along with their densities (finite and non-zero on the entire space)
-/
variable {d : ℕ}

variable [MeasurableSpace (Vector ℝ d)] [MeasureSpace (Vector ℝ d)] [MeasureSpace ℝ]

variable (μ π ν : Measure (Vector ℝ d)) (dμ dπ : (Vector ℝ d) 0)

/-
  μ << π << ν, they both admit density w.r.t. ν.
-/
variable (_h1 : absolutely_continuous μ π) (_h2 : absolutely_continuous π ν)
example : absolutely_continuous μ ν := absolutely_continuous_trans _h1 _h2

variable (hμ : is_density μ ν dμ) (hπ : is_density π ν dπ) (mdμ : Measurable dμ) (mdπ : Measurable dπ) (hdμ : x, dμ x  0  dμ x  ) (hdπ : x, dπ x  0  dπ x  )




variable [IsProbabilityMeasure μ] [IsProbabilityMeasure π]

variable (h_m_set :  (s : Set (Vector ℝ d)), MeasurableSet s)


/-
  We define a RKHS of ((Vector ℝ d) → ℝ) functions.
-/
variable (H₀ : Set ((Vector ℝ d)  ℝ)) [NormedAddCommGroup ((Vector ℝ d)  ℝ)] [InnerProductSpace ℝ ((Vector ℝ d)  ℝ)]

/- The kernel function -/
variable (k : (Vector ℝ d)  (Vector ℝ d)  ℝ) (h_k : ( (x : (Vector ℝ d)), k x  H₀)  ( (x : (Vector ℝ d)), (fun y  k y x)  H₀))

variable (h_kernel : is_kernel H₀ k) (h_kernel_positive : positive_definite_kernel μ k)

/- We define the product RKHS as a space of function on ℕ → (Vector ℝ d) to ℝ (vector-valued function in our Lean formalism). A function belongs to such a RKHS if f = (f_1, ..., f_d) and ∀ 1 ≤ i ≤ d, fᵢ ∈ H₀. -/
variable {H : Set (ℕ  (Vector ℝ d)  ℝ)} [NormedAddCommGroup (ℕ  (Vector ℝ d)  ℝ)] [InnerProductSpace ℝ (ℕ  (Vector ℝ d)  ℝ)]

/-===============================KERNEL STEIN DISCREPANCY===============================-/
/-
Here, we prove that KSD(μ | π) is a valid discrepancy measure, that the Stein log Sobolev inequality holds, and the exponential convergence of SVGD.
-/

/- dk : x ↦ i ↦ y ↦ ∂xⁱ k(x, y) -/
variable (dk : (Vector ℝ d)  (Vector ℝ d)  ℝ)

/- d_log_π : i ↦ x ↦ ∂xⁱ log (μ(x) / π(x)) -/
variable (d_log_π : ℕ  (Vector ℝ d)  ℝ)

/- Definition of the steepest direction ϕ -/
variable (ϕ : ℕ  (Vector ℝ d)  ℝ) (hϕ : ϕ  H) (dϕ : ℕ  (Vector ℝ d)  ℝ)

variable (h_is_ϕ : is_phi μ k dk d_log_π ϕ)

/- We will use this assumption only when the function is trivially integrable (e.g. derivative of integrable functions). -/
variable (is_integrable_H₀ :  (f : Vector ℝ d  ℝ), Integrable f μ)

/-
d_log_π_μ : i ↦ x ↦ ∂xⁱ log (π(x) / μ(x))
-/
variable (d_log_π_μ : ℕ  (Vector ℝ d)  ℝ)

/-
Simple derivative rule: if the derivative is 0 ∀x, then the function is constant.
-/
variable (hd_log_π_μ : (x, i, d_log_π_μ i x = 0)  ( c,  x, log (dμ x / dπ x) = c))

/-
dπ' : i ↦ x ↦ ∂xⁱ π(x)
-/
variable (dπ' : ℕ  (Vector ℝ d)  ℝ)

/-
Simple derivative rule: ∂xⁱ log (π(x)) * π(x) = ∂xⁱ π(x).
-/
variable (hπ' : x, i, ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x)


variable [Norm (Vector ℝ d)]
/--
  Stein class of measure. f is in the Stein class of μ if, ∀i ∈ range (d + 1), lim_(‖x‖ → ∞) μ(x) * ϕ(x)ⁱ = 0.
-/
def SteinClass (f : ℕ  (Vector ℝ d)  ℝ) (dμ : (Vector ℝ d) 0) :=  x, tends_to_infty (fun (x : Vector ℝ d)  x)  i, ENNReal.toReal (dμ x) * f i x = 0


/-
  Kernel Stein Discrepancy
-/
variable (KSD : Measure (Vector ℝ d)  Measure (Vector ℝ d)  ℝ)

/--
KSD(μ | π) = ⟪∇log π/μ, Pμ ∇log π/μ⟫_L²(μ). We assume here that KSD is also equal to ∫ x, ∑ l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) ∂μ.
-/
def is_ksd := KSD μ π = ( x in Set.univ, ( x' in Set.univ, ( i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x') μ) μ)  (KSD μ π =  x,  l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) μ)

/-
  KSD(μ | π) is originally defined as ‖ϕ^⋆‖²_H, it is therefore non-negative.
-/
variable (ksd_nn : 0  KSD μ π)

/-
  ϕ is in the Stein class of π
-/
variable (hstein : SteinClass ϕ dπ)

/--
  We show that, if ϕ is in the Stein class of π, KSD is a valid discrepancy measure i.e. μ = π ↔ KSD(μ | π) = 0.
-/
lemma KSD_is_valid_discrepancy (hksd : is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD) : μ = π  KSD μ π = 0 :=

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

μ = π KSD μ π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

mp
μ = π KSD μ π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
KSD μ π = 0 μ = π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

μ = π KSD μ π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

mp
μ = π KSD μ π = 0
-- μ = π ↦ KSD(μ | π) = 0.
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π

mp
KSD μ π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

mp
μ = π KSD μ π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
KSD μ π = 0 μ = π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π

mp
KSD μ π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π

mp
(x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) μ = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π

mp
(x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) μ = 0
-- ∑ i, f i + g i = ∑ i, f i + ∑ i, g i.
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x

mp
(x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) μ = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

mp
μ = π KSD μ π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
KSD μ π = 0 μ = π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x

mp
(x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) μ = 0
[Meta.Tactic.simp.rewrite] split_sum:1000, l in Finset.range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) ==> l in Finset.range (d + 1), d_log_π l x * ϕ l x + l in Finset.range (d + 1), dϕ l x
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x

mp
(x : Vector ℝ d), l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x μ = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x

mp
(x : Vector ℝ d), l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x μ = 0
-- Split the integral of sum into sum of integral.
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x

mp
(x : Vector ℝ d), l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x μ = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

mp
μ = π KSD μ π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
KSD μ π = 0 μ = π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x

mp
(x : Vector ℝ d), l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x μ = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

mp
μ = π KSD μ π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
KSD μ π = 0 μ = π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x

mp
(x : Vector ℝ d), l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x μ = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x

mp
(a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ + (a : Vector ℝ d), l in range (d + 1), dϕ l a μ = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x

mp
(a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ + (a : Vector ℝ d), l in range (d + 1), dϕ l a μ = 0
-- Make the `Set.univ` appears for using the density later.
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x

mp
(a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ + (a : Vector ℝ d), l in range (d + 1), dϕ l a μ = 0

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x

(a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
[Meta.Tactic.simp.rewrite] @MeasureTheory.Measure.restrict_univ:1000, MeasureTheory.Measure.restrict μ Set.univ ==> μ [Meta.Tactic.simp.rewrite] @eq_self:1000, (a : Vector ℝ d), l in Finset.range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d), l in Finset.range (d + 1), d_log_π l a * ϕ l a μ ==> True

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

mp
μ = π KSD μ π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
KSD μ π = 0 μ = π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ

mp
(a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ + (a : Vector ℝ d), l in range (d + 1), dϕ l a μ = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ

mp
(a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ + (a : Vector ℝ d), l in range (d + 1), dϕ l a μ = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ

mp
(a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ + (a : Vector ℝ d), l in range (d + 1), dϕ l a μ = 0
-- Replace μ by π in the integration.
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ

mp
(a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ + (a : Vector ℝ d), l in range (d + 1), dϕ l a μ = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ

mp
(a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a π + (a : Vector ℝ d), l in range (d + 1), dϕ l a π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ

mp
(a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a π + (a : Vector ℝ d), l in range (d + 1), dϕ l a π = 0
-- Replace by its density.
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ

mp
(a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a π + (a : Vector ℝ d), l in range (d + 1), dϕ l a π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ

mp
(x : Vector ℝ d) in Set.univ, ENNReal.toReal (dπ x) * l in range (d + 1), d_log_π l x * ϕ l x ν + (a : Vector ℝ d), l in range (d + 1), dϕ l a π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ

mp
(x : Vector ℝ d) in Set.univ, ENNReal.toReal (dπ x) * l in range (d + 1), d_log_π l x * ϕ l x ν + (a : Vector ℝ d), l in range (d + 1), dϕ l a π = 0
-- Get ENNReal.toReal (dπ x) in the sum (a * ∑ b = ∑ b * a).
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ

mp
(x : Vector ℝ d) in Set.univ, ENNReal.toReal (dπ x) * l in range (d + 1), d_log_π l x * ϕ l x ν + (a : Vector ℝ d), l in range (d + 1), dϕ l a π = 0

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ

(x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ

(x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ

(x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
mul_dist_sum: (a : ℝ) (f : ℕ ℝ), ( i in range (d + 1), f i) * a = i in range (d + 1), f i * a

(x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ

(x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
mul_dist_sum: (a : ℝ) (f : ℕ ℝ), ( i in range (d + 1), f i) * a = i in range (d + 1), f i * a
x: Vector ℝ d

ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ

(x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
mul_dist_sum: (a : ℝ) (f : ℕ ℝ), ( i in range (d + 1), f i) * a = i in range (d + 1), f i * a
x: Vector ℝ d

ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
mul_dist_sum: (a : ℝ) (f : ℕ ℝ), ( i in range (d + 1), f i) * a = i in range (d + 1), f i * a
x: Vector ℝ d

( l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l) * ENNReal.toReal (dπ x) = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
mul_dist_sum: (a : ℝ) (f : ℕ ℝ), ( i in range (d + 1), f i) * a = i in range (d + 1), f i * a
x: Vector ℝ d

( l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l) * ENNReal.toReal (dπ x) = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ

(x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

mp
μ = π KSD μ π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
KSD μ π = 0 μ = π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
mul_dist: (x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)

mp
(x : Vector ℝ d) in Set.univ, ENNReal.toReal (dπ x) * l in range (d + 1), d_log_π l x * ϕ l x ν + (a : Vector ℝ d), l in range (d + 1), dϕ l a π = 0
[Meta.Tactic.simp.rewrite] mul_dist:1000, ENNReal.toReal (dπ x) * l in Finset.range (d + 1), d_log_π l x * ϕ l x ==> l in Finset.range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
mul_dist: (x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)

mp
(x : Vector ℝ d) in Set.univ, x_1 in range (d + 1), d_log_π x_1 x * ϕ x_1 x * ENNReal.toReal (dπ x) ν + (a : Vector ℝ d), l in range (d + 1), dϕ l a π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
mul_dist: (x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)

mp
(x : Vector ℝ d) in Set.univ, x_1 in range (d + 1), d_log_π x_1 x * ϕ x_1 x * ENNReal.toReal (dπ x) ν + (a : Vector ℝ d), l in range (d + 1), dϕ l a π = 0
-- Make the product ENNReal.toReal (dπ x) * d_log_π i x appears to use the log derivative rule.
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
mul_dist: (x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
mul_comm: (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x

mp
(x : Vector ℝ d) in Set.univ, x_1 in range (d + 1), d_log_π x_1 x * ϕ x_1 x * ENNReal.toReal (dπ x) ν + (a : Vector ℝ d), l in range (d + 1), dϕ l a π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

mp
μ = π KSD μ π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
KSD μ π = 0 μ = π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
mul_dist: (x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
mul_comm: (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x

mp
(x : Vector ℝ d) in Set.univ, x_1 in range (d + 1), d_log_π x_1 x * ϕ x_1 x * ENNReal.toReal (dπ x) ν + (a : Vector ℝ d), l in range (d + 1), dϕ l a π = 0
[Meta.Tactic.simp.rewrite] mul_comm:1000, d_log_π x x * ϕ x x * ENNReal.toReal (dπ x) ==> ENNReal.toReal (dπ x) * d_log_π x x * ϕ x x
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
mul_dist: (x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
mul_comm: (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x

mp
(x : Vector ℝ d) in Set.univ, x_1 in range (d + 1), d_log_π x_1 x * ϕ x_1 x * ENNReal.toReal (dπ x) ν + (a : Vector ℝ d), l in range (d + 1), dϕ l a π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
mul_dist: (x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
mul_comm: (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x

mp
(x : Vector ℝ d) in Set.univ, x_1 in range (d + 1), ENNReal.toReal (dπ x) * d_log_π x_1 x * ϕ x_1 x ν + (a : Vector ℝ d), l in range (d + 1), dϕ l a π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
mul_dist: (x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
mul_comm: (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x

mp
(x : Vector ℝ d) in Set.univ, x_1 in range (d + 1), d_log_π x_1 x * ϕ x_1 x * ENNReal.toReal (dπ x) ν + (a : Vector ℝ d), l in range (d + 1), dϕ l a π = 0
[Meta.Tactic.simp.rewrite] hπ':1000, ENNReal.toReal (dπ x) * d_log_π x x ==> dπ' x x
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
mul_dist: (x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
mul_comm: (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x

mp
(x : Vector ℝ d) in Set.univ, x_1 in range (d + 1), dπ' x_1 x * ϕ x_1 x ν + (a : Vector ℝ d), l in range (d + 1), dϕ l a π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
mul_dist: (x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
mul_comm: (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x

mp
(x : Vector ℝ d) in Set.univ, x_1 in range (d + 1), dπ' x_1 x * ϕ x_1 x ν + (a : Vector ℝ d), l in range (d + 1), dϕ l a π = 0
-- Make the `Set.univ` appears to use the density.
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
mul_dist: (x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
mul_comm: (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x

mp
(x : Vector ℝ d) in Set.univ, x_1 in range (d + 1), dπ' x_1 x * ϕ x_1 x ν + (a : Vector ℝ d), l in range (d + 1), dϕ l a π = 0

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
mul_dist: (x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
mul_comm: (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x

(a : Vector ℝ d), l in range (d + 1), dϕ l a π = (a : Vector ℝ d) in Set.univ, l in range (d + 1), dϕ l a π
[Meta.Tactic.simp.rewrite] @MeasureTheory.Measure.restrict_univ:1000, MeasureTheory.Measure.restrict π Set.univ ==> π [Meta.Tactic.simp.rewrite] @eq_self:1000, (a : Vector ℝ d), l in Finset.range (d + 1), dϕ l a π = (a : Vector ℝ d), l in Finset.range (d + 1), dϕ l a π ==> True

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

mp
μ = π KSD μ π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
KSD μ π = 0 μ = π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ✝: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
mul_dist: (x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
mul_comm: (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x
int_univ: (a : Vector ℝ d), l in range (d + 1), dϕ l a π = (a : Vector ℝ d) in Set.univ, l in range (d + 1), dϕ l a π

mp
(x : Vector ℝ d) in Set.univ, x_1 in range (d + 1), dπ' x_1 x * ϕ x_1 x ν + (a : Vector ℝ d), l in range (d + 1), dϕ l a π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ✝: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
mul_dist: (x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
mul_comm: (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x
int_univ: (a : Vector ℝ d), l in range (d + 1), dϕ l a π = (a : Vector ℝ d) in Set.univ, l in range (d + 1), dϕ l a π

mp
(x : Vector ℝ d) in Set.univ, x_1 in range (d + 1), dπ' x_1 x * ϕ x_1 x ν + (a : Vector ℝ d) in Set.univ, l in range (d + 1), dϕ l a π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ✝: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
mul_dist: (x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
mul_comm: (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x
int_univ: (a : Vector ℝ d), l in range (d + 1), dϕ l a π = (a : Vector ℝ d) in Set.univ, l in range (d + 1), dϕ l a π

mp
(x : Vector ℝ d) in Set.univ, x_1 in range (d + 1), dπ' x_1 x * ϕ x_1 x ν + (a : Vector ℝ d) in Set.univ, l in range (d + 1), dϕ l a π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

mp
μ = π KSD μ π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
KSD μ π = 0 μ = π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ✝: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
mul_dist: (x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
mul_comm: (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x
int_univ: (a : Vector ℝ d), l in range (d + 1), dϕ l a π = (a : Vector ℝ d) in Set.univ, l in range (d + 1), dϕ l a π

mp
(x : Vector ℝ d) in Set.univ, x_1 in range (d + 1), dπ' x_1 x * ϕ x_1 x ν + (a : Vector ℝ d) in Set.univ, l in range (d + 1), dϕ l a π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ✝: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
mul_dist: (x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
mul_comm: (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x
int_univ: (a : Vector ℝ d), l in range (d + 1), dϕ l a π = (a : Vector ℝ d) in Set.univ, l in range (d + 1), dϕ l a π

mp
(x : Vector ℝ d) in Set.univ, x_1 in range (d + 1), dπ' x_1 x * ϕ x_1 x ν + (x : Vector ℝ d) in Set.univ, ENNReal.toReal (dπ x) * l in range (d + 1), dϕ l x ν = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ✝: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
mul_dist: (x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
mul_comm: (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x
int_univ: (a : Vector ℝ d), l in range (d + 1), dϕ l a π = (a : Vector ℝ d) in Set.univ, l in range (d + 1), dϕ l a π

mp
(x : Vector ℝ d) in Set.univ, x_1 in range (d + 1), dπ' x_1 x * ϕ x_1 x ν + (x : Vector ℝ d) in Set.univ, ENNReal.toReal (dπ x) * l in range (d + 1), dϕ l x ν = 0
-- Use the integration by parts on the right-hand side integral.
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ✝: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
mul_dist: (x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
mul_comm: (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x
int_univ: (a : Vector ℝ d), l in range (d + 1), dϕ l a π = (a : Vector ℝ d) in Set.univ, l in range (d + 1), dϕ l a π

mp
(x : Vector ℝ d) in Set.univ, x_1 in range (d + 1), dπ' x_1 x * ϕ x_1 x ν + (x : Vector ℝ d) in Set.univ, ENNReal.toReal (dπ x) * l in range (d + 1), dϕ l x ν = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ✝: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
mul_dist: (x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
mul_comm: (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x
int_univ: (a : Vector ℝ d), l in range (d + 1), dϕ l a π = (a : Vector ℝ d) in Set.univ, l in range (d + 1), dϕ l a π

mp
(x : Vector ℝ d) in Set.univ, x_1 in range (d + 1), dπ' x_1 x * ϕ x_1 x ν + -∫ (x : Vector ℝ d) in Set.univ, i in range (d + 1), dπ' i x * ϕ i x ν = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
split_sum: (x : Vector ℝ d), l in range (d + 1), (d_log_π l x * ϕ l x + dϕ l x) = l in range (d + 1), d_log_π l x * ϕ l x + l in range (d + 1), dϕ l x
h1: Integrable fun x => l in range (d + 1), d_log_π l x * ϕ l x
h2: Integrable fun x => l in range (d + 1), dϕ l x
int_univ✝: (a : Vector ℝ d), l in range (d + 1), d_log_π l a * ϕ l a μ = (a : Vector ℝ d) in Set.univ, l in range (d + 1), d_log_π l a * ϕ l a μ
mul_dist: (x : Vector ℝ d), ENNReal.toReal (dπ x) * l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l = l in range (d + 1), (fun l => d_log_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
mul_comm: (x : Vector ℝ d) (i : ℕ), d_log_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_log_π i x * ϕ i x
int_univ: (a : Vector ℝ d), l in range (d + 1), dϕ l a π = (a : Vector ℝ d) in Set.univ, l in range (d + 1), dϕ l a π

mp
(x : Vector ℝ d) in Set.univ, x_1 in range (d + 1), dπ' x_1 x * ϕ x_1 x ν + -∫ (x : Vector ℝ d) in Set.univ, i in range (d + 1), dπ' i x * ϕ i x ν = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

mp
μ = π KSD μ π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
KSD μ π = 0 μ = π
[Meta.Tactic.simp.rewrite] @MeasureTheory.Measure.restrict_univ:1000, MeasureTheory.Measure.restrict ν Set.univ ==> ν [Meta.Tactic.simp.rewrite] @add_right_neg:1000, (x : Vector ℝ d), x_1 in Finset.range (d + 1), dπ' x_1 x * ϕ x_1 x ν + -∫ (x : Vector ℝ d), x_1 in Finset.range (d + 1), dπ' x_1 x * ϕ x_1 x ν ==> 0 [Meta.Tactic.simp.rewrite] @eq_self:1000, 0 = 0 ==> True

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

mpr
KSD μ π = 0 μ = π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

μ = π KSD μ π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

mpr
KSD μ π = 0 μ = π
-- KSD(μ | π) = 0 ↦ μ = π.
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: KSD μ π = 0

mpr
μ = π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

mpr
KSD μ π = 0 μ = π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: KSD μ π = 0

mpr
μ = π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0

mpr
μ = π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0

mpr
μ = π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0

mpr
μ = π
-- We use the fact that the kernel is positive-definite that implies that d_log_π_μ = 0.
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0

mpr
μ = π
-- Simple derivative rule: ∂x f x = 0 → f x = c
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
hd_log_π_μ: c, (x : Vector ℝ d), log (dμ x / dπ x) = c

mpr
μ = π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

mpr
KSD μ π = 0 μ = π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c

mpr.intro
μ = π
-- We show that, since dμ x / dπ x ≠ 0 and finite, dμ x = ENNReal.ofReal (Real.exp c) * dπ x.
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c

mpr.intro
μ = π

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c

(x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c

(x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c

(x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
x: Vector ℝ d

dμ x = ENNReal.ofReal (Real.exp c) * dπ x
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c

(x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c

dμ x = ENNReal.ofReal (Real.exp c) * dπ x
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c

(x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c

dμ x = ENNReal.ofReal (Real.exp c) * dπ x

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c

dμ x / dπ x 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c

dμ x / dπ x 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c

dμ x / dπ x 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c
frac_pos: 0 < dμ x / dπ x

dμ x / dπ x 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c

dμ x / dπ x 0

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c

(x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0

dμ x = ENNReal.ofReal (Real.exp c) * dπ x

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0

dμ x / dπ x
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0

dμ x / dπ x
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0

dμ x / dπ x
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝¹: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h✝: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0
h: dμ x / dπ x =

False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0

dμ x / dπ x
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝¹: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h✝: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0
h: dμ x / dπ x =

False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝¹: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h✝: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0
h: dμ x 0 dπ x = 0 dμ x = dπ x

False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝¹: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h✝: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0
h: dμ x 0 dπ x = 0 dμ x = dπ x

False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝¹: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h✝: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0
h: dμ x 0 dπ x = 0 dμ x = dπ x

False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0

dμ x / dπ x
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝¹: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h✝: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0
h: dμ x 0 dπ x = 0 dμ x = dπ x

False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0
hp: dμ x 0 dπ x = 0

inl
False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0
hp: dμ x 0 dπ x = 0

inl
False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0
hp: dμ x 0 dπ x = 0

inl
False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0
hp: dμ x 0 dπ x = 0

inl
False
Warning: unused variable `hpl` [linter.unusedVariables]
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0
hp: dμ x 0 dπ x = 0

inl
False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0
hpl: dμ x 0
hpr: dπ x = 0

inl.intro
False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0
hp: dμ x 0 dπ x = 0

inl
False

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝¹: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h✝: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0
h: dμ x 0 dπ x = 0 dμ x = dπ x

False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0
hq: dμ x = dπ x

inr
False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0
hq: dμ x = dπ x

inr
False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0
hq: dμ x = dπ x

inr
False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0
hq: dμ x = dπ x

inr
False
Warning: unused variable `hqr` [linter.unusedVariables]
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0
hq: dμ x = dπ x

inr
False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0
hql: dμ x =
hqr: dπ x

inr.intro
False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0
hq: dμ x = dπ x

inr
False

Goals accomplished! 🐙

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c

(x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
x: Vector ℝ d
h: log (dμ x / dπ x) = c
frac_neq_zero: dμ x / dπ x 0
frac_finite: dμ x / dπ x
cancel_log_exp: dμ x / dπ x = ENNReal.ofReal (Real.exp c)

dμ x = ENNReal.ofReal (Real.exp c) * dπ x
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c

(x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
[Meta.Tactic.simp.rewrite] cancel_log_exp:1000, ENNReal.ofReal (Real.exp c) ==> dμ x / dπ x [Meta.Tactic.simp.rewrite] @ENNReal.div_eq_inv_mul:1000, dμ x / dπ x ==> (dπ x)⁻¹ * dμ x [Meta.Tactic.simp.rewrite] mul_right_comm (dπ x)⁻¹ (dμ x) (dπ x):1000, (dπ x)⁻¹ * dμ x * dπ x ==> (dπ x)⁻¹ * dπ x * dμ x [Meta.Tactic.simp.rewrite] ENNReal.inv_mul_cancel (hdπ x).left (hdπ x).right:1000, (dπ x)⁻¹ * dπ x ==> 1 [Meta.Tactic.simp.rewrite] @one_mul:1000, 1 * dμ x ==> dμ x [Meta.Tactic.simp.rewrite] @eq_self:1000, dμ x = dμ x ==> True

Goals accomplished! 🐙

Goals accomplished! 🐙
-- We show by cases that ENNReal.ofReal (Real.exp c) = 1. If it is ≠ 1, this implies a contradiction as dμ x = ENNReal.ofReal (Real.exp c) * dπ x and ∫⁻ x, dμ x ∂ν = 1.
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x

mpr.intro
μ = π

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x

ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x

ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x

ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) = 1

pos
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ¬ENNReal.ofReal (Real.exp c) = 1
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x

ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) = 1

pos
ENNReal.ofReal (Real.exp c) = 1

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ¬ENNReal.ofReal (Real.exp c) = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x

ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ¬ENNReal.ofReal (Real.exp c) = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ¬ENNReal.ofReal (Real.exp c) = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ¬ENNReal.ofReal (Real.exp c) = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1

neg
ENNReal.ofReal (Real.exp c) = 1
Warning: unused variable `x` [linter.unusedVariables]
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1

neg
ENNReal.ofReal (Real.exp c) = 1

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1

∫⁻ (x : Vector ℝ d) in Set.univ, 1 μ = 1
[Meta.Tactic.simp.rewrite] @MeasureTheory.Measure.restrict_univ:1000, MeasureTheory.Measure.restrict μ Set.univ ==> μ [Meta.Tactic.simp.rewrite] @MeasureTheory.lintegral_const:1000, ∫⁻ (x : Vector ℝ d), 1 μ ==> 1 * ↑↑μ Set.univ [Meta.Tactic.simp.rewrite] @MeasureTheory.measure_univ:1000, ↑↑μ Set.univ ==> 1 [Meta.Tactic.simp.rewrite] @mul_one:1000, 1 * 1 ==> 1 [Meta.Tactic.simp.rewrite] @eq_self:1000, 1 = 1 ==> True

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ¬ENNReal.ofReal (Real.exp c) = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, 1 μ = 1

neg
ENNReal.ofReal (Real.exp c) = 1
Warning: unused variable `x` [linter.unusedVariables]
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, 1 μ = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, 1 μ = 1

neg
ENNReal.ofReal (Real.exp c) = 1

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, 1 μ = 1

∫⁻ (x : Vector ℝ d) in Set.univ, 1 π = 1
[Meta.Tactic.simp.rewrite] @MeasureTheory.Measure.restrict_univ:1000, MeasureTheory.Measure.restrict π Set.univ ==> π [Meta.Tactic.simp.rewrite] @MeasureTheory.lintegral_const:1000, ∫⁻ (x : Vector ℝ d), 1 π ==> 1 * ↑↑π Set.univ [Meta.Tactic.simp.rewrite] @MeasureTheory.measure_univ:1000, ↑↑π Set.univ ==> 1 [Meta.Tactic.simp.rewrite] @mul_one:1000, 1 * 1 ==> 1 [Meta.Tactic.simp.rewrite] @eq_self:1000, 1 = 1 ==> True

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ¬ENNReal.ofReal (Real.exp c) = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, 1 μ = 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, 1 π = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, 1 μ = 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, 1 π = 1

neg
ENNReal.ofReal (Real.exp c) = 1
Warning: unused variable `x` [linter.unusedVariables]
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, 1 μ = 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, 1 π = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, dμ x * 1 ν = 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, 1 π = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, dμ x * 1 ν = 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, 1 π = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, dμ x * 1 ν = 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, 1 π = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ¬ENNReal.ofReal (Real.exp c) = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, dμ x * 1 ν = 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, 1 π = 1

neg
ENNReal.ofReal (Real.exp c) = 1
[Meta.Tactic.simp.rewrite] dμ_propor:1000, dμ x ==> ENNReal.ofReal (Real.exp c) * dπ x
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, 1 π = 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x * 1 ν = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, 1 π = 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x * 1 ν = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ¬ENNReal.ofReal (Real.exp c) = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, 1 π = 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x * 1 ν = 1

neg
ENNReal.ofReal (Real.exp c) = 1
[Meta.Tactic.simp.rewrite] @mul_one:1000, ENNReal.ofReal (Real.exp c) * dπ x * 1 ==> ENNReal.ofReal (Real.exp c) * dπ x
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, 1 π = 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ν = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, 1 π = 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ν = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ¬ENNReal.ofReal (Real.exp c) = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, 1 π = 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ν = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, 1 π = 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ν = 1

neg
ENNReal.ofReal (Real.exp c) = 1
Warning: unused variable `x` [linter.unusedVariables]
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, 1 π = 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ν = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x * 1 ν = 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ν = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x * 1 ν = 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ν = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x * 1 ν = 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ν = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ¬ENNReal.ofReal (Real.exp c) = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x * 1 ν = 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ν = 1

neg
ENNReal.ofReal (Real.exp c) = 1
[Meta.Tactic.simp.rewrite] @mul_one:1000, dπ x * 1 ==> dπ x
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ν = 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x ν = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ν = 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x ν = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ¬ENNReal.ofReal (Real.exp c) = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ν = 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x ν = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_μ: ENNReal.ofReal (Real.exp c) * ∫⁻ (a : Vector ℝ d) in Set.univ, dπ a ν = 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x ν = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ν = 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x ν = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_μ: ENNReal.ofReal (Real.exp c) * 1 = 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x ν = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_μ: ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ν = 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x ν = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_μ: ENNReal.ofReal (Real.exp c) = 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x ν = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_μ: ENNReal.ofReal (Real.exp c) = 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x ν = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_μ: ENNReal.ofReal (Real.exp c) = 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x ν = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ¬ENNReal.ofReal (Real.exp c) = 1

neg
ENNReal.ofReal (Real.exp c) = 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ENNReal.ofReal (Real.exp c) 1
univ_eq_one_μ: ENNReal.ofReal (Real.exp c) = 1
univ_eq_one_π: ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x ν = 1

neg.h
False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
hc: ¬ENNReal.ofReal (Real.exp c) = 1

neg
ENNReal.ofReal (Real.exp c) = 1

Goals accomplished! 🐙

Goals accomplished! 🐙

Goals accomplished! 🐙
-- We rewrite μ = π as ∀s, ∫⁻ x in s, dμ ∂ν = ∀s, ∫⁻ x in s, dπ ∂ν and use dμ = 1 * dπ.
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1

mpr.intro
μ = π
[Meta.Tactic.simp.rewrite] exp_c_eq_one:1000, ENNReal.ofReal (Real.exp c) ==> 1
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1

mpr.intro
μ = π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1
dμ_propor: (x : Vector ℝ d), dμ x = 1 * dπ x

mpr.intro
μ = π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
dμ_propor: (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1

mpr.intro
μ = π
[Meta.Tactic.simp.rewrite] @one_mul:1000, 1 * dπ x ==> dπ x
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1
dμ_propor: (x : Vector ℝ d), dμ x = dπ x

mpr.intro
μ = π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1
dμ_propor: (x : Vector ℝ d), dμ x = dπ x

mpr.intro
μ = π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

mpr
KSD μ π = 0 μ = π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1
dμ_propor: (x : Vector ℝ d), dμ x = dπ x
s: Set (Vector ℝ d)
_hs: MeasurableSet s

mpr.intro.h
↑↑μ s = ↑↑π s
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

mpr
KSD μ π = 0 μ = π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1
dμ_propor: (x : Vector ℝ d), dμ x = dπ x
s: Set (Vector ℝ d)
_hs: MeasurableSet s

mpr.intro.h
↑↑μ s = ↑↑π s
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1
dμ_propor: (x : Vector ℝ d), dμ x = dπ x
s: Set (Vector ℝ d)
_hs: MeasurableSet s

mpr.intro.h
∫⁻ (x : Vector ℝ d) in s, 1 μ = ↑↑π s
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1
dμ_propor: (x : Vector ℝ d), dμ x = dπ x
s: Set (Vector ℝ d)
_hs: MeasurableSet s

mpr.intro.h
↑↑μ s = ↑↑π s
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1
dμ_propor: (x : Vector ℝ d), dμ x = dπ x
s: Set (Vector ℝ d)
_hs: MeasurableSet s

mpr.intro.h
∫⁻ (x : Vector ℝ d) in s, 1 μ = ∫⁻ (x : Vector ℝ d) in s, 1 π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1
dμ_propor: (x : Vector ℝ d), dμ x = dπ x
s: Set (Vector ℝ d)
_hs: MeasurableSet s

mpr.intro.h
∫⁻ (x : Vector ℝ d) in s, 1 μ = ∫⁻ (x : Vector ℝ d) in s, 1 π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

mpr
KSD μ π = 0 μ = π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1
dμ_propor: (x : Vector ℝ d), dμ x = dπ x
s: Set (Vector ℝ d)
_hs: MeasurableSet s

mpr.intro.h
∫⁻ (x : Vector ℝ d) in s, 1 μ = ∫⁻ (x : Vector ℝ d) in s, 1 π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1
dμ_propor: (x : Vector ℝ d), dμ x = dπ x
s: Set (Vector ℝ d)
_hs: MeasurableSet s

mpr.intro.h
∫⁻ (x : Vector ℝ d) in s, dμ x * 1 ν = ∫⁻ (x : Vector ℝ d) in s, 1 π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1
dμ_propor: (x : Vector ℝ d), dμ x = dπ x
s: Set (Vector ℝ d)
_hs: MeasurableSet s

mpr.intro.h
∫⁻ (x : Vector ℝ d) in s, 1 μ = ∫⁻ (x : Vector ℝ d) in s, 1 π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1
dμ_propor: (x : Vector ℝ d), dμ x = dπ x
s: Set (Vector ℝ d)
_hs: MeasurableSet s

mpr.intro.h
∫⁻ (x : Vector ℝ d) in s, dμ x * 1 ν = ∫⁻ (x : Vector ℝ d) in s, dπ x * 1 ν
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1
dμ_propor: (x : Vector ℝ d), dμ x = dπ x
s: Set (Vector ℝ d)
_hs: MeasurableSet s

mpr.intro.h
∫⁻ (x : Vector ℝ d) in s, dμ x * 1 ν = ∫⁻ (x : Vector ℝ d) in s, dπ x * 1 ν
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

mpr
KSD μ π = 0 μ = π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1
dμ_propor: (x : Vector ℝ d), dμ x = dπ x
s: Set (Vector ℝ d)
_hs: MeasurableSet s

mpr.intro.h
∫⁻ (x : Vector ℝ d) in s, dμ x * 1 ν = ∫⁻ (x : Vector ℝ d) in s, dπ x * 1 ν
[Meta.Tactic.simp.rewrite] @mul_one:1000, dμ x * 1 ==> dμ x [Meta.Tactic.simp.rewrite] @mul_one:1000, dπ x * 1 ==> dπ x
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1
dμ_propor: (x : Vector ℝ d), dμ x = dπ x
s: Set (Vector ℝ d)
_hs: MeasurableSet s

mpr.intro.h
∫⁻ (x : Vector ℝ d) in s, dμ x * 1 ν = ∫⁻ (x : Vector ℝ d) in s, dπ x * 1 ν
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1
dμ_propor: (x : Vector ℝ d), dμ x = dπ x
s: Set (Vector ℝ d)
_hs: MeasurableSet s

mpr.intro.h
∫⁻ (x : Vector ℝ d) in s, dμ x ν = ∫⁻ (x : Vector ℝ d) in s, dπ x ν
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ, dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h✝: (x : Vector ℝ d) in Set.univ, (x' : Vector ℝ d) in Set.univ, i in range (d + 1), d_log_π_μ i x * k x x' * d_log_π_μ i x' μ μ = 0
d_log_π_μ_eq_0: (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0
c:
h: (x : Vector ℝ d), log (dμ x / dπ x) = c
exp_c_eq_one: ENNReal.ofReal (Real.exp c) = 1
dμ_propor: (x : Vector ℝ d), dμ x = dπ x
s: Set (Vector ℝ d)
_hs: MeasurableSet s

mpr.intro.h
∫⁻ (x : Vector ℝ d) in s, dμ x * 1 ν = ∫⁻ (x : Vector ℝ d) in s, dπ x * 1 ν
[Meta.Tactic.simp.rewrite] dμ_propor:1000, dμ x ==> dπ x [Meta.Tactic.simp.rewrite] @eq_self:1000, ∫⁻ (x : Vector ℝ d) in s, dπ x ν = ∫⁻ (x : Vector ℝ d) in s, dπ x ν ==> True

Goals accomplished! 🐙

Goals accomplished! 🐙

Goals accomplished! 🐙
/- Basic proprieties of KL. -/ variable (hkl_eq : μ = π KL μ dμ dπ = 0) (hkl_diff : μ π 0 < KL μ dμ dπ) /-- We show that μ ≠ π → 0 < KSD μ π (trivial using *KSD_is_valid_discrepancy*). -/ lemma μ_neq_π_imp_ksd_nn (hksd : is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD) : μ π 0 < KSD μ π :=

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

μ π 0 < KSD μ π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π

0 < KSD μ π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

μ π 0 < KSD μ π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
h2: ¬0 < KSD μ π

False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

μ π 0 < KSD μ π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
h2: KSD μ π 0

False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

μ π 0 < KSD μ π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
h2: KSD μ π 0
split_le: KSD μ π < 0 KSD μ π = 0

False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

μ π 0 < KSD μ π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
h2: KSD μ π 0
split_le: KSD μ π < 0 KSD μ π = 0

False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
h2: KSD μ π 0
lt: KSD μ π < 0

inl
False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
h2: KSD μ π 0
lt: KSD μ π < 0

inl
False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
h2: KSD μ π 0
lt: KSD μ π < 0

inl
False

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
h2: KSD μ π 0
split_le: KSD μ π < 0 KSD μ π = 0

False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
h2: KSD μ π 0
eq: KSD μ π = 0

inr
False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
h2: KSD μ π 0
eq: KSD μ π = 0

inr
False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
h2: KSD μ π 0
eq: KSD μ π = 0

inr
False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
h2: KSD μ π 0
eq: KSD μ π = 0
μ_eq_π: μ = π

inr
False
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
h2: KSD μ π 0
eq: KSD μ π = 0

inr
False

Goals accomplished! 🐙

Goals accomplished! 🐙
/-- We show that it exists a finite and positive θ such that KL(μ || π) ≤ (1 / (2θ)) * KSD(μ | π) -/ theorem Stein_log_Sobolev (hksd : is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD) : θ > 0, (θ ) (KL μ dμ dπ (1 / (2*θ)) * ENNReal.ofReal (KSD μ π)) :=

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

θ > 0, θ KL μ dμ dπ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π

pos
θ > 0, θ KL μ dμ dπ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: ¬μ = π
θ > 0, θ KL μ dμ dπ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

θ > 0, θ KL μ dμ dπ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π

pos
θ > 0, θ KL μ dμ dπ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
-- μ = π → KSD μ π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π

pos
θ > 0, θ KL μ dμ dπ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π

pos
θ > 0, θ KL μ dμ dπ 1 / (2 * θ) * ENNReal.ofReal 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π

pos
θ > 0, θ KL μ dμ dπ 1 / (2 * θ) * ENNReal.ofReal 0
-- μ = π → KL μ π = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π

pos
θ > 0, θ KL μ dμ dπ 1 / (2 * θ) * ENNReal.ofReal 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π

pos
θ > 0, θ 0 1 / (2 * θ) * ENNReal.ofReal 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π

pos
θ > 0, θ 0 1 / (2 * θ) * ENNReal.ofReal 0
-- Use any θ > 0 ∧ θ ≠ ∞
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π

h
1 > 0 1 0 1 / (2 * 1) * ENNReal.ofReal 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π

pos
θ > 0, θ KL μ dμ dπ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: ¬μ = π
θ > 0, θ KL μ dμ dπ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π

h.left
1 > 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π
1 0 1 / (2 * 1) * ENNReal.ofReal 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π

pos
θ > 0, θ KL μ dμ dπ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: ¬μ = π
θ > 0, θ KL μ dμ dπ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π

h.left
1 > 0
[Meta.Tactic.simp.rewrite] @gt_iff_lt:1000, 1 > 0 ==> 0 < 1 [Meta.Tactic.simp.rewrite] @zero_lt_one:1000, 0 < 1 ==> True

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π

h.right
1 0 1 / (2 * 1) * ENNReal.ofReal 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ = π

pos
θ > 0, θ KL μ dμ dπ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: ¬μ = π
θ > 0, θ KL μ dμ dπ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
[Meta.Tactic.simp.rewrite] @mul_one:1000, 2 * 1 ==> 2 [Meta.Tactic.simp.rewrite] @one_div:1000, 1 / 2 ==> 2⁻¹ [Meta.Tactic.simp.rewrite] ENNReal.ofReal_zero:1000, ENNReal.ofReal 0 ==> 0 [Meta.Tactic.simp.rewrite] @mul_zero:1000, 2⁻¹ * 0 ==> 0 [Meta.Tactic.simp.rewrite] @le_refl:1000, 0 0 ==> True [Meta.Tactic.simp.rewrite] and_self:1000, True True ==> True

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: ¬μ = π

neg
θ > 0, θ KL μ dμ dπ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD

θ > 0, θ KL μ dμ dπ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: ¬μ = π

neg
θ > 0, θ KL μ dμ dπ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
-- μ ≠ π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π

neg
θ > 0, θ KL μ dμ dπ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
-- Let θ = KSD(μ | π) / (2 KL(μ || π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π

h
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) > 0 ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) KL μ dμ dπ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: ¬μ = π

neg
θ > 0, θ KL μ dμ dπ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π

h.left
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) > 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) KL μ dμ dπ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: ¬μ = π

neg
θ > 0, θ KL μ dμ dπ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π

h.left
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) > 0
-- We show that 0 < KSD(μ | π) / (2 KL(μ || π) by showing that 0 < KSD(μ | π) and 2 KL(μ || π) ≠ ∞ (as both are non-negative).
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π

h.left
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) > 0

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π

0 < KSD μ π 2 * KL μ dμ dπ 0 < ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ)
[Meta.Tactic.simp.rewrite] @ne_eq:1000, 2 * KL μ dμ dπ ==> ¬2 * KL μ dμ dπ = [Meta.Tactic.simp.rewrite] @ENNReal.div_pos_iff:1000, 0 < ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) ==> ENNReal.ofReal (KSD μ π) 0 2 * KL μ dμ dπ [Meta.Tactic.simp.rewrite] @ne_eq:1000, ENNReal.ofReal (KSD μ π) 0 ==> ¬ENNReal.ofReal (KSD μ π) = 0 [Meta.Tactic.simp.rewrite] @ENNReal.ofReal_eq_zero:1000, ENNReal.ofReal (KSD μ π) = 0 ==> KSD μ π 0 [Meta.Tactic.simp.rewrite] @not_le:1000, ¬KSD μ π 0 ==> 0 < KSD μ π [Meta.Tactic.simp.rewrite] @imp_self:1000, 0 < KSD μ π ¬2 * KL μ dμ dπ = 0 < KSD μ π ¬2 * KL μ dμ dπ = ==> True

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π

h.left
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) > 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) KL μ dμ dπ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
imp_lt: 0 < KSD μ π 2 * KL μ dμ dπ 0 < ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ)

h.left
0 < KSD μ π 2 * KL μ dμ dπ
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π

h.left
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) > 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) KL μ dμ dπ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
imp_lt: 0 < KSD μ π 2 * KL μ dμ dπ 0 < ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ)

h.left.left
0 < KSD μ π
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
imp_lt: 0 < KSD μ π 2 * KL μ dμ dπ 0 < ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ)
2 * KL μ dμ dπ
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π

h.left
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) > 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) KL μ dμ dπ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
imp_lt: 0 < KSD μ π 2 * KL μ dμ dπ 0 < ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ)

h.left.left
0 < KSD μ π
-- We use *μ_neq_π_imp_ksd_nn* as μ ≠ π.

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
imp_lt: 0 < KSD μ π 2 * KL μ dμ dπ 0 < ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ)

h.left.right
2 * KL μ dμ dπ
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π

h.left
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) > 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) KL μ dμ dπ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
imp_lt: 0 < KSD μ π 2 * KL μ dμ dπ 0 < ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ)

h.left.right
2 * KL μ dμ dπ
-- KL is finite (in our framework, as μ << π << ν).
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
imp_lt: 0 < KSD μ π 2 * KL μ dμ dπ 0 < ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ)

h.left.right
2 * KL μ dμ dπ

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
imp_lt: 0 < KSD μ π 2 * KL μ dμ dπ 0 < ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ)

2

Goals accomplished! 🐙

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π

h.right
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) KL μ dμ dπ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: ¬μ = π

neg
θ > 0, θ KL μ dμ dπ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π

h.right
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) KL μ dμ dπ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
-- μ ≠ π → KL(μ || π) ≠ 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0

h.right
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) KL μ dμ dπ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π

h.right
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) KL μ dμ dπ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0

h.right.left
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
KL μ dμ dπ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π

h.right
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) KL μ dμ dπ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0

h.right.left
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ)
-- We show that KSD(μ | π) / (2 KL(μ || π) ≠ ∞ by showing that KSD(μ | π) ≠ ∞ and 2 (KL(μ || π))⁻¹ ≠ ∞ (as both are non-negative).
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
div_as_inv: ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹

h.right.left
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0

h.right.left
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
KL μ dμ dπ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
div_as_inv: ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹

h.right.left
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
div_as_inv: ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹

h.right.left
ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
div_as_inv: ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹

h.right.left
ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0

h.right.left
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
KL μ dμ dπ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
div_as_inv: ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹
enn_KSD_finite: ENNReal.ofReal (KSD μ π)

h.right.left
ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹
-- We show that (KL(μ || π))⁻¹ ≠ ∞ as KL(μ || π) ≠ 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
div_as_inv: ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹
enn_KSD_finite: ENNReal.ofReal (KSD μ π)

h.right.left
ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
div_as_inv: ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹
enn_KSD_finite: ENNReal.ofReal (KSD μ π)

(2 * KL μ dμ dπ)⁻¹
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
div_as_inv: ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹
enn_KSD_finite: ENNReal.ofReal (KSD μ π)

(2 * KL μ dμ dπ)⁻¹
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
div_as_inv: ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹
enn_KSD_finite: ENNReal.ofReal (KSD μ π)

(2 * KL μ dμ dπ)⁻¹
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
div_as_inv: ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹
enn_KSD_finite: ENNReal.ofReal (KSD μ π)

(2 * KL μ dμ dπ)⁻¹

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
div_as_inv: ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹
enn_KSD_finite: ENNReal.ofReal (KSD μ π)

2 * KL μ dμ dπ 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
div_as_inv: ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹
enn_KSD_finite: ENNReal.ofReal (KSD μ π)

2 * KL μ dμ dπ 0
[Meta.Tactic.simp.rewrite] @ne_eq:1000, 2 * KL μ dμ dπ 0 ==> ¬2 * KL μ dμ dπ = 0 [Meta.Tactic.simp.rewrite] @mul_eq_zero:1000, 2 * KL μ dμ dπ = 0 ==> 2 = 0 KL μ dμ dπ = 0 [Meta.Tactic.simp.rewrite] @OfNat.ofNat_ne_zero:1000, 2 = 0 ==> False [Meta.Tactic.simp.rewrite] false_or:1000, False KL μ dμ dπ = 0 ==> KL μ dμ dπ = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
div_as_inv: ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹
enn_KSD_finite: ENNReal.ofReal (KSD μ π)

¬KL μ dμ dπ = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
div_as_inv: ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹
enn_KSD_finite: ENNReal.ofReal (KSD μ π)

¬KL μ dμ dπ = 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
div_as_inv: ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹
enn_KSD_finite: ENNReal.ofReal (KSD μ π)

2 * KL μ dμ dπ 0

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
div_as_inv: ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹
enn_KSD_finite: ENNReal.ofReal (KSD μ π)

(2 * KL μ dμ dπ)⁻¹

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0

h.right.left
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
KL μ dμ dπ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0

h.right.right
KL μ dμ dπ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π

h.right
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) KL μ dμ dπ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0

h.right.right
KL μ dμ dπ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
-- We show that, under non-zero and finite conditions, a ≤ (1 / (2 * (b / (2 * a)))) * b (in fact, a = (1 / (2 * (b / (2 * a)))) * b).
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0

h.right.right
KL μ dμ dπ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0

(a b : ℝ0), a 0 a b 0 b a 1 / (2 * (b / (2 * a))) * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0

(a b : ℝ0), a 0 a b 0 b a 1 / (2 * (b / (2 * a))) * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0

(a b : ℝ0), a 0 a b 0 b a 1 / (2 * (b / (2 * a))) * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b

a 1 / (2 * (b / (2 * a))) * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0

(a b : ℝ0), a 0 a b 0 b a 1 / (2 * (b / (2 * a))) * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b

a 1 / (2 * (b / (2 * a))) * b

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b

1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹
[Meta.Tactic.simp.rewrite] @one_div:1000, 1 / (2 * (b / (2 * a))) ==> (2 * (b / (2 * a)))⁻¹ [Meta.Tactic.simp.rewrite] @eq_self:1000, (2 * (b / (2 * a)))⁻¹ = (2 * (b / (2 * a)))⁻¹ ==> True

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0

(a b : ℝ0), a 0 a b 0 b a 1 / (2 * (b / (2 * a))) * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

a 1 / (2 * (b / (2 * a))) * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

a (2 * (b / (2 * a)))⁻¹ * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

a (2 * (b / (2 * a)))⁻¹ * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0

(a b : ℝ0), a 0 a b 0 b a 1 / (2 * (b / (2 * a))) * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

a (2 * (b / (2 * a)))⁻¹ * b

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * (b / (2 * a)))⁻¹ * b = a
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * (b / (2 * a)))⁻¹ * b = a
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * (b / (2 * a)))⁻¹ * b = a
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * (b / (2 * a)))⁻¹ * b = a

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * (b / (2 * a)))⁻¹ * b = (2 * (b / (2 * a)))⁻¹ * b

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * (b / (2 * a)))⁻¹ * b = a

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * (b / (2 * a)))⁻¹ * b = (2 * (b * (2 * a)⁻¹))⁻¹ * b

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * (b / (2 * a)))⁻¹ * b = a

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * (b * (2 * a)⁻¹))⁻¹ * b = (2 * b * (2 * a)⁻¹)⁻¹ * b
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.mul_assoc_rev:1000, 2 * (b * (2 * a)⁻¹) ==> 2 * b * (2 * a)⁻¹ [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1 ==> 1 [Meta.Tactic.simp.rewrite] @pow_one:1000, (2 * b * (2 * a)⁻¹)⁻¹ ^ 1 ==> (2 * b * (2 * a)⁻¹)⁻¹ [Meta.Tactic.simp.rewrite] @pow_one:1000, b ^ 1 ==> b [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1 ==> 1 [Meta.Tactic.simp.rewrite] @mul_one:1000, b * 1 ==> b [Meta.Tactic.simp.rewrite] @add_zero:1000, (2 * b * (2 * a)⁻¹)⁻¹ * b + 0 ==> (2 * b * (2 * a)⁻¹)⁻¹ * b [Meta.Tactic.simp.rewrite] @pow_one:1000, (2 * b * (2 * a)⁻¹)⁻¹ ^ 1 ==> (2 * b * (2 * a)⁻¹)⁻¹ [Meta.Tactic.simp.rewrite] @mul_one:1000, (2 * b * (2 * a)⁻¹)⁻¹ * 1 ==> (2 * b * (2 * a)⁻¹)⁻¹ [Meta.Tactic.simp.rewrite] @add_zero:1000, b * (2 * b * (2 * a)⁻¹)⁻¹ + 0 ==> b * (2 * b * (2 * a)⁻¹)⁻¹ [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1 ==> 1 [Meta.Tactic.simp.rewrite] @pow_one:1000, a ^ 1 ==> a [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2 ==> 2 [Meta.Tactic.simp.rewrite] @add_zero:1000, a * 2 + 0 ==> a * 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1 ==> 1 [Meta.Tactic.simp.rewrite] @pow_one:1000, a ^ 1 ==> a [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2 ==> 2 [Meta.Tactic.simp.rewrite] @add_zero:1000, a * 2 + 0 ==> a * 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1 ==> 1 [Meta.Tactic.simp.rewrite] @pow_one:1000, b ^ 1 ==> b [Meta.Tactic.simp.rewrite] @pow_one:1000, (a * 2)⁻¹ ^ 1 ==> (a * 2)⁻¹ [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2 ==> 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.mul_assoc_rev:1000, b * ((a * 2)⁻¹ * 2) ==> b * (a * 2)⁻¹ * 2 [Meta.Tactic.simp.rewrite] @add_zero:1000, b * (a * 2)⁻¹ * 2 + 0 ==> b * (a * 2)⁻¹ * 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1 ==> 1 [Meta.Tactic.simp.rewrite] @pow_one:1000, a ^ 1 ==> a [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2 ==> 2 [Meta.Tactic.simp.rewrite] @add_zero:1000, a * 2 + 0 ==> a * 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1 ==> 1 [Meta.Tactic.simp.rewrite] @pow_one:1000, b ^ 1 ==> b [Meta.Tactic.simp.rewrite] @pow_one:1000, (a * 2)⁻¹ ^ 1 ==> (a * 2)⁻¹ [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2 ==> 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.mul_assoc_rev:1000, b * ((a * 2)⁻¹ * 2) ==> b * (a * 2)⁻¹ * 2 [Meta.Tactic.simp.rewrite] @add_zero:1000, b * (a * 2)⁻¹ * 2 + 0 ==> b * (a * 2)⁻¹ * 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1 ==> 1 [Meta.Tactic.simp.rewrite] @pow_one:1000, b ^ 1 ==> b [Meta.Tactic.simp.rewrite] @pow_one:1000, (b * (a * 2)⁻¹ * 2)⁻¹ ^ 1 ==> (b * (a * 2)⁻¹ * 2)⁻¹ [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1 ==> 1 [Meta.Tactic.simp.rewrite] @mul_one:1000, (b * (a * 2)⁻¹ * 2)⁻¹ * 1 ==> (b * (a * 2)⁻¹ * 2)⁻¹ [Meta.Tactic.simp.rewrite] @add_zero:1000, b * (b * (a * 2)⁻¹ * 2)⁻¹ + 0 ==> b * (b * (a * 2)⁻¹ * 2)⁻¹ [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1 ==> 1 [Meta.Tactic.simp.rewrite] @pow_one:1000, a ^ 1 ==> a [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2 ==> 2 [Meta.Tactic.simp.rewrite] @add_zero:1000, a * 2 + 0 ==> a * 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1 ==> 1 [Meta.Tactic.simp.rewrite] @pow_one:1000, a ^ 1 ==> a [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2 ==> 2 [Meta.Tactic.simp.rewrite] @add_zero:1000, a * 2 + 0 ==> a * 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1 ==> 1 [Meta.Tactic.simp.rewrite] @pow_one:1000, b ^ 1 ==> b [Meta.Tactic.simp.rewrite] @pow_one:1000, (a * 2)⁻¹ ^ 1 ==> (a * 2)⁻¹ [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2 ==> 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.mul_assoc_rev:1000, b * ((a * 2)⁻¹ * 2) ==> b * (a * 2)⁻¹ * 2 [Meta.Tactic.simp.rewrite] @add_zero:1000, b * (a * 2)⁻¹ * 2 + 0 ==> b * (a * 2)⁻¹ * 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1 ==> 1 [Meta.Tactic.simp.rewrite] @pow_one:1000, a ^ 1 ==> a [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2 ==> 2 [Meta.Tactic.simp.rewrite] @add_zero:1000, a * 2 + 0 ==> a * 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1 ==> 1 [Meta.Tactic.simp.rewrite] @pow_one:1000, b ^ 1 ==> b [Meta.Tactic.simp.rewrite] @pow_one:1000, (a * 2)⁻¹ ^ 1 ==> (a * 2)⁻¹ [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2 ==> 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.mul_assoc_rev:1000, b * ((a * 2)⁻¹ * 2) ==> b * (a * 2)⁻¹ * 2 [Meta.Tactic.simp.rewrite] @add_zero:1000, b * (a * 2)⁻¹ * 2 + 0 ==> b * (a * 2)⁻¹ * 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1 ==> 1 [Meta.Tactic.simp.rewrite] @pow_one:1000, b ^ 1 ==> b [Meta.Tactic.simp.rewrite] @pow_one:1000, (b * (a * 2)⁻¹ * 2)⁻¹ ^ 1 ==> (b * (a * 2)⁻¹ * 2)⁻¹ [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1 ==> 1 [Meta.Tactic.simp.rewrite] @mul_one:1000, (b * (a * 2)⁻¹ * 2)⁻¹ * 1 ==> (b * (a * 2)⁻¹ * 2)⁻¹ [Meta.Tactic.simp.rewrite] @add_zero:1000, b * (b * (a * 2)⁻¹ * 2)⁻¹ + 0 ==> b * (b * (a * 2)⁻¹ * 2)⁻¹ [Meta.Tactic.simp.rewrite] @eq_self:1000, b * (b * (a * 2)⁻¹ * 2)⁻¹ = b * (b * (a * 2)⁻¹ * 2)⁻¹ ==> True Try this: ring_nf

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * (b / (2 * a)))⁻¹ * b = a

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * b * (2 * a)⁻¹)⁻¹ * b = (2 * 2⁻¹ * a⁻¹ * b)⁻¹ * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * b * (2 * a)⁻¹)⁻¹ * b = (2 * 2⁻¹ * a⁻¹ * b)⁻¹ * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * b * (2 * a)⁻¹)⁻¹ * b = (2 * 2⁻¹ * a⁻¹ * b)⁻¹ * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * b * (2 * a)⁻¹)⁻¹ * b = (2 * 2⁻¹ * a⁻¹ * b)⁻¹ * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * b * (2 * a)⁻¹)⁻¹ * b = (2 * 2⁻¹ * a⁻¹ * b)⁻¹ * b

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

2 0 a
[Meta.Tactic.simp.rewrite] @ne_eq:1000, 2 0 ==> ¬2 = 0 [Meta.Tactic.simp.rewrite] @OfNat.ofNat_ne_zero:1000, 2 = 0 ==> False [Meta.Tactic.simp.rewrite] not_false_eq_true:1000, ¬False ==> True [Meta.Tactic.simp.rewrite] @ne_eq:1000, a ==> ¬a = [Meta.Tactic.simp.rewrite] true_or:1000, True ¬a = ==> True

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * b * (2⁻¹ * a⁻¹))⁻¹ * b = (2 * 2⁻¹ * a⁻¹ * b)⁻¹ * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * b * (2⁻¹ * a⁻¹))⁻¹ * b = (2 * 2⁻¹ * a⁻¹ * b)⁻¹ * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * b * (2 * a)⁻¹)⁻¹ * b = (2 * 2⁻¹ * a⁻¹ * b)⁻¹ * b
[Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.mul_assoc_rev:1000, 2 * b * (2⁻¹ * a⁻¹) ==> 2 * b * 2⁻¹ * a⁻¹ [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1 ==> 1 [Meta.Tactic.simp.rewrite] @pow_one:1000, (2 * b * 2⁻¹ * a⁻¹)⁻¹ ^ 1 ==> (2 * b * 2⁻¹ * a⁻¹)⁻¹ [Meta.Tactic.simp.rewrite] @pow_one:1000, b ^ 1 ==> b [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1 ==> 1 [Meta.Tactic.simp.rewrite] @mul_one:1000, b * 1 ==> b [Meta.Tactic.simp.rewrite] @add_zero:1000, (2 * b * 2⁻¹ * a⁻¹)⁻¹ * b + 0 ==> (2 * b * 2⁻¹ * a⁻¹)⁻¹ * b [Meta.Tactic.simp.rewrite] @pow_one:1000, (2 * 2⁻¹ * a⁻¹ * b)⁻¹ ^ 1 ==> (2 * 2⁻¹ * a⁻¹ * b)⁻¹ [Meta.Tactic.simp.rewrite] @mul_one:1000, (2 * 2⁻¹ * a⁻¹ * b)⁻¹ * 1 ==> (2 * 2⁻¹ * a⁻¹ * b)⁻¹ [Meta.Tactic.simp.rewrite] @add_zero:1000, b * (2 * 2⁻¹ * a⁻¹ * b)⁻¹ + 0 ==> b * (2 * 2⁻¹ * a⁻¹ * b)⁻¹ [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2 ==> 2 [Meta.Tactic.simp.rewrite] @add_zero:1000, 2 + 0 ==> 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2 ==> 2 [Meta.Tactic.simp.rewrite] @add_zero:1000, 2 + 0 ==> 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1 ==> 1 [Meta.Tactic.simp.rewrite] @pow_one:1000, b ^ 1 ==> b [Meta.Tactic.simp.rewrite] @pow_one:1000, 2⁻¹ ^ 1 ==> 2⁻¹ [Meta.Tactic.simp.rewrite] @pow_one:1000, a⁻¹ ^ 1 ==> a⁻¹ [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2 ==> 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.mul_assoc_rev:1000, 2⁻¹ * (a⁻¹ * 2) ==> 2⁻¹ * a⁻¹ * 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.mul_assoc_rev:1000, b * (2⁻¹ * a⁻¹ * 2) ==> b * (2⁻¹ * a⁻¹) * 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.mul_assoc_rev:1000, b * (2⁻¹ * a⁻¹) ==> b * 2⁻¹ * a⁻¹ [Meta.Tactic.simp.rewrite] @add_zero:1000, b * 2⁻¹ * a⁻¹ * 2 + 0 ==> b * 2⁻¹ * a⁻¹ * 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2 ==> 2 [Meta.Tactic.simp.rewrite] @add_zero:1000, 2 + 0 ==> 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1 ==> 1 [Meta.Tactic.simp.rewrite] @pow_one:1000, b ^ 1 ==> b [Meta.Tactic.simp.rewrite] @pow_one:1000, 2⁻¹ ^ 1 ==> 2⁻¹ [Meta.Tactic.simp.rewrite] @pow_one:1000, a⁻¹ ^ 1 ==> a⁻¹ [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2 ==> 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.mul_assoc_rev:1000, 2⁻¹ * (a⁻¹ * 2) ==> 2⁻¹ * a⁻¹ * 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.mul_assoc_rev:1000, b * (2⁻¹ * a⁻¹ * 2) ==> b * (2⁻¹ * a⁻¹) * 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.mul_assoc_rev:1000, b * (2⁻¹ * a⁻¹) ==> b * 2⁻¹ * a⁻¹ [Meta.Tactic.simp.rewrite] @add_zero:1000, b * 2⁻¹ * a⁻¹ * 2 + 0 ==> b * 2⁻¹ * a⁻¹ * 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1 ==> 1 [Meta.Tactic.simp.rewrite] @pow_one:1000, b ^ 1 ==> b [Meta.Tactic.simp.rewrite] @pow_one:1000, (b * 2⁻¹ * a⁻¹ * 2)⁻¹ ^ 1 ==> (b * 2⁻¹ * a⁻¹ * 2)⁻¹ [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1 ==> 1 [Meta.Tactic.simp.rewrite] @mul_one:1000, (b * 2⁻¹ * a⁻¹ * 2)⁻¹ * 1 ==> (b * 2⁻¹ * a⁻¹ * 2)⁻¹ [Meta.Tactic.simp.rewrite] @add_zero:1000, b * (b * 2⁻¹ * a⁻¹ * 2)⁻¹ + 0 ==> b * (b * 2⁻¹ * a⁻¹ * 2)⁻¹ [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2 ==> 2 [Meta.Tactic.simp.rewrite] @add_zero:1000, 2 + 0 ==> 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2 ==> 2 [Meta.Tactic.simp.rewrite] @add_zero:1000, 2 + 0 ==> 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1 ==> 1 [Meta.Tactic.simp.rewrite] @pow_one:1000, b ^ 1 ==> b [Meta.Tactic.simp.rewrite] @pow_one:1000, 2⁻¹ ^ 1 ==> 2⁻¹ [Meta.Tactic.simp.rewrite] @pow_one:1000, a⁻¹ ^ 1 ==> a⁻¹ [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2 ==> 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.mul_assoc_rev:1000, 2⁻¹ * (a⁻¹ * 2) ==> 2⁻¹ * a⁻¹ * 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.mul_assoc_rev:1000, b * (2⁻¹ * a⁻¹ * 2) ==> b * (2⁻¹ * a⁻¹) * 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.mul_assoc_rev:1000, b * (2⁻¹ * a⁻¹) ==> b * 2⁻¹ * a⁻¹ [Meta.Tactic.simp.rewrite] @add_zero:1000, b * 2⁻¹ * a⁻¹ * 2 + 0 ==> b * 2⁻¹ * a⁻¹ * 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2 ==> 2 [Meta.Tactic.simp.rewrite] @add_zero:1000, 2 + 0 ==> 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1 ==> 1 [Meta.Tactic.simp.rewrite] @pow_one:1000, b ^ 1 ==> b [Meta.Tactic.simp.rewrite] @pow_one:1000, 2⁻¹ ^ 1 ==> 2⁻¹ [Meta.Tactic.simp.rewrite] @pow_one:1000, a⁻¹ ^ 1 ==> a⁻¹ [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_2:1000, Nat.rawCast 2 ==> 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.mul_assoc_rev:1000, 2⁻¹ * (a⁻¹ * 2) ==> 2⁻¹ * a⁻¹ * 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.mul_assoc_rev:1000, b * (2⁻¹ * a⁻¹ * 2) ==> b * (2⁻¹ * a⁻¹) * 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.mul_assoc_rev:1000, b * (2⁻¹ * a⁻¹) ==> b * 2⁻¹ * a⁻¹ [Meta.Tactic.simp.rewrite] @add_zero:1000, b * 2⁻¹ * a⁻¹ * 2 + 0 ==> b * 2⁻¹ * a⁻¹ * 2 [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1 ==> 1 [Meta.Tactic.simp.rewrite] @pow_one:1000, b ^ 1 ==> b [Meta.Tactic.simp.rewrite] @pow_one:1000, (b * 2⁻¹ * a⁻¹ * 2)⁻¹ ^ 1 ==> (b * 2⁻¹ * a⁻¹ * 2)⁻¹ [Meta.Tactic.simp.rewrite] @Mathlib.Tactic.RingNF.nat_rawCast_1:1000, Nat.rawCast 1 ==> 1 [Meta.Tactic.simp.rewrite] @mul_one:1000, (b * 2⁻¹ * a⁻¹ * 2)⁻¹ * 1 ==> (b * 2⁻¹ * a⁻¹ * 2)⁻¹ [Meta.Tactic.simp.rewrite] @add_zero:1000, b * (b * 2⁻¹ * a⁻¹ * 2)⁻¹ + 0 ==> b * (b * 2⁻¹ * a⁻¹ * 2)⁻¹ [Meta.Tactic.simp.rewrite] @eq_self:1000, b * (b * 2⁻¹ * a⁻¹ * 2)⁻¹ = b * (b * 2⁻¹ * a⁻¹ * 2)⁻¹ ==> True Try this: ring_nf

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * (b / (2 * a)))⁻¹ * b = a

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * 2⁻¹ * a⁻¹ * b)⁻¹ * b = (a⁻¹ * b)⁻¹ * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * 2⁻¹ * a⁻¹ * b)⁻¹ * b = (a⁻¹ * b)⁻¹ * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * 2⁻¹ * a⁻¹ * b)⁻¹ * b = (a⁻¹ * b)⁻¹ * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * 2⁻¹ * a⁻¹ * b)⁻¹ * b = (a⁻¹ * b)⁻¹ * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * 2⁻¹ * a⁻¹ * b)⁻¹ * b = (a⁻¹ * b)⁻¹ * b

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

2 0
[Meta.Tactic.simp.rewrite] @ne_eq:1000, 2 0 ==> ¬2 = 0 [Meta.Tactic.simp.rewrite] @OfNat.ofNat_ne_zero:1000, 2 = 0 ==> False [Meta.Tactic.simp.rewrite] not_false_eq_true:1000, ¬False ==> True

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * 2⁻¹ * a⁻¹ * b)⁻¹ * b = (a⁻¹ * b)⁻¹ * b

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

2

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(1 * a⁻¹ * b)⁻¹ * b = (a⁻¹ * b)⁻¹ * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * 2⁻¹ * a⁻¹ * b)⁻¹ * b = (a⁻¹ * b)⁻¹ * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(a⁻¹ * b)⁻¹ * b = (a⁻¹ * b)⁻¹ * b

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * (b / (2 * a)))⁻¹ * b = a

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(a⁻¹ * b)⁻¹ * b = a * b⁻¹ * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(a⁻¹ * b)⁻¹ * b = a * b⁻¹ * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(a⁻¹ * b)⁻¹ * b = a * b⁻¹ * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹
t: a⁻¹ 0

(a⁻¹ * b)⁻¹ * b = a * b⁻¹ * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(a⁻¹ * b)⁻¹ * b = a * b⁻¹ * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹
t: a⁻¹ 0

(a⁻¹ * b)⁻¹ * b = a * b⁻¹ * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹
t: a⁻¹ 0

a⁻¹⁻¹ * b⁻¹ * b = a * b⁻¹ * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹
t: a⁻¹ 0

a⁻¹⁻¹ * b⁻¹ * b = a * b⁻¹ * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(a⁻¹ * b)⁻¹ * b = a * b⁻¹ * b
[Meta.Tactic.simp.rewrite] @inv_inv:1000, a⁻¹⁻¹ ==> a [Meta.Tactic.simp.rewrite] @eq_self:1000, a * b⁻¹ * b = a * b⁻¹ * b ==> True

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * (b / (2 * a)))⁻¹ * b = a

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

a * b⁻¹ * b = a * (b⁻¹ * b)

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

(2 * (b / (2 * a)))⁻¹ * b = a

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

a * (b⁻¹ * b) = a
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

a * (b⁻¹ * b) = a
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

a * (b⁻¹ * b) = a
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

a * (b⁻¹ * b) = a
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

a * 1 = a
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

a * (b⁻¹ * b) = a
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹

a = a

Goals accomplished! 🐙

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0

(a b : ℝ0), a 0 a b 0 b a 1 / (2 * (b / (2 * a))) * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹
eq: (2 * (b / (2 * a)))⁻¹ * b = a

a (2 * (b / (2 * a)))⁻¹ * b
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
a, b: 0
h0a: a 0
hta: a
h0b: b 0
htb: b
simpl: 1 / (2 * (b / (2 * a))) = (2 * (b / (2 * a)))⁻¹
eq: (2 * (b / (2 * a)))⁻¹ * b = a

a a

Goals accomplished! 🐙

Goals accomplished! 🐙
-- As μ ≠ π, 0 < KSD(μ | π) and thus, KSD(μ | π) ≠ 0.
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
calculation: (a b : ℝ0), a 0 a b 0 b a 1 / (2 * (b / (2 * a))) * b

h.right.right
KL μ dμ dπ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)

Goals accomplished! 🐙
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
calculation: (a b : ℝ0), a 0 a b 0 b a 1 / (2 * (b / (2 * a))) * b

ENNReal.ofReal (KSD μ π) 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
calculation: (a b : ℝ0), a 0 a b 0 b a 1 / (2 * (b / (2 * a))) * b

ENNReal.ofReal (KSD μ π) 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
calculation: (a b : ℝ0), a 0 a b 0 b a 1 / (2 * (b / (2 * a))) * b

ENNReal.ofReal (KSD μ π) 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
calculation: (a b : ℝ0), a 0 a b 0 b a 1 / (2 * (b / (2 * a))) * b
KSD_ge_0: 0 < KSD μ π

ENNReal.ofReal (KSD μ π) 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
calculation: (a b : ℝ0), a 0 a b 0 b a 1 / (2 * (b / (2 * a))) * b

ENNReal.ofReal (KSD μ π) 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
calculation: (a b : ℝ0), a 0 a b 0 b a 1 / (2 * (b / (2 * a))) * b
KSD_ge_0: 0 < KSD μ π
enn_KSD_ge_0: 0 < ENNReal.ofReal (KSD μ π)

ENNReal.ofReal (KSD μ π) 0
d:
inst✝⁹: MeasurableSpace (Vector ℝ d)
inst✝⁸: MeasureSpace (Vector ℝ d)
inst✝⁷: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝⁶: IsProbabilityMeasure μ
inst✝⁵: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁴: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝³: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝²: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝¹: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
hksd: is_ksd μ π k d_log_π ϕ dϕ d_log_π_μ KSD
h: μ π
KL_neq_0: KL μ dμ dπ 0
calculation: (a b : ℝ0), a 0 a b 0 b a 1 / (2 * (b / (2 * a))) * b

ENNReal.ofReal (KSD μ π) 0

Goals accomplished! 🐙

Goals accomplished! 🐙
-- Use the previous calculation with a := KL(μ || π), b := KSD(μ | π).

Goals accomplished! 🐙

Goals accomplished! 🐙

Goals accomplished! 🐙

Goals accomplished! 🐙
/- In this sub-section, we define the flow of measures μ_t: μ_t : ℝ≥0 → Measure (Vector ℝ d) t ↦ T_t#μ, where T_t is the trajectories associated with ϕ(μ_t t), the steepest direction to update μ_t t for minimizing ∂t KL(μ_t t || π). We also define everything that we need to use previous results with each measures given by μ_t. -/ variable (μ_t : ℝ0 Measure (Vector ℝ d)) (dμ_t : ℝ0 (Vector ℝ d 0)) (hμ_t : t, is_density (μ_t t) ν (dμ_t t)) (h_prob : t, IsProbabilityMeasure (μ_t t)) variable (hdμ_t :t, (x : Vector ℝ d), dμ_t t x 0 dμ_t t x ) /- d_KL_t : t ↦ ∂t KL(μ_t t || π) -/ variable (d_KL_t : ℝ0 ℝ) variable (d_log_μ_t_π : ℝ0 (Vector ℝ d) ℝ) variable (hd_log_μ_t_π : t, (x, i, d_log_μ_t_π t i x = 0) ( c, x, log (dμ_t t x / dπ x) = c)) variable (hkl_eq_t : t, μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0) (hkl_diff_t : t, μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ) variable (h_kernel_positive_t : t, positive_definite_kernel (μ_t t) k) variable (is_integrable_H₀_t : t, (f : Vector ℝ d ℝ), Integrable f (μ_t t)) variable (ksd_nn_t : t, 0 KSD (μ_t t) π) /- Our definition of the Gronwall's lemma. -/ variable [MeasureSpace ℝ0] [NormedAddCommGroup ℝ0] [NormedSpace ℝ ℝ0] [LocallyFiniteOrder ℝ0] variable (gronwall : (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( s in Icc 0 t, ψ s)) /- We assume here that ∂t KL(μ_t t || π) = - KSD(μ_t t | π) (showed in the paper). -/ variable (dkl_ksd : t, d_KL_t t - KSD (μ_t t) π) /- As showed in the paper (and reminded above), t ↦ KL(μ_t t || π) is decreasing. -/ variable (kl_decreasing : t, t', t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ) (kl_finite : t, KL (μ_t t) (dμ_t t) dπ ) /-- If t ↦ KL(μ_t t || π) is bounded from below by a strictly positive constant, it means that μ_t cannot be made arbitrary close to π and thus that t ↦ KSD(μ_t | π) can be bounded by a striclty positive constant. Admitted here; we plan on formally show it in the future. -/ lemma
Warning: declaration uses 'sorry'
(α : ℝ0) (hα : 0 < α) (hkl : t, α < KL (μ_t t) (dμ_t t) dπ) : (β : ℝ0), (t, β < ENNReal.ofReal (KSD (μ_t t) π)) (0 < β) ) :=

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
α: 0
: 0 < α
hkl: (t : ℝ0), α < KL (μ_t t) (dμ_t t) dπ

β, ( (t : ℝ0), β < ENNReal.ofReal (KSD (μ_t t) π)) 0 < β β

Goals accomplished! 🐙
/-- Squeeze theorem: ∀t, 0 ≤ KL(μ_t t || π) ≤ f(t) ∧ lim_(t → ∞) f(t) = 0 → lim_(t → ∞) KL(μ_t t || π) = 0. -/ lemma
Warning: declaration uses 'sorry'
(f : ℝ0 0) (h : limit f 0) : (t>0, KL (μ_t t) (dμ_t t) dπ f t) limit (fun t KL (μ_t t) (dμ_t t) dπ) 0 :=

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
f: 0 0
h: limit f 0

( t > 0, KL (μ_t t) (dμ_t t) dπ f t) limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0

Goals accomplished! 🐙
/-- We show the convergence of SVGD i.e. lim_(t → ∞) KL (μ_t || π) = 0. -/ theorem convergence_SVGD (log_sobolev : t, KL (μ_t t) (dμ_t t) dπ (1/(2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * (KL (μ_t t) (dμ_t t) dπ)) ) )) * ENNReal.ofReal (KSD (μ_t t) π) ) : limit (fun t KL (μ_t t) (dμ_t t) dπ) 0 :=

Goals accomplished! 🐙
-- As t ↦ KL (μ_t || π) is decreasing and bound from below, it admits a limit l ≥ 0.
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)

limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)

(x : ℝ0), 0 (fun t => KL (μ_t t) (dμ_t t) dπ) x
[Meta.Tactic.simp.rewrite] @zero_le:1000, 0 KL (μ_t x) (dμ_t x) dπ ==> True [Meta.Tactic.simp.rewrite] @forall_const:1000, ℝ0 True ==> True

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
admits_limit: α, limit (fun t => KL (μ_t t) (dμ_t t) dπ) α 0 α (x : ℝ0), α < (fun t => KL (μ_t t) (dμ_t t) dπ) x

limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)

limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x

intro.intro.intro
limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
-- We proceed by cases on the value of l. If l = 0, the proof is finished. Otherwise, we proceed by contradiction by showing that l ≠ 0 → l = 0.
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 = l

pos
limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: ¬0 = l
limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)

limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 = l

pos
limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 = l

pos
limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: ¬0 = l
limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 = l

pos
limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 = l

pos
limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 = l

pos
limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: ¬0 = l

neg
limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)

limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: ¬0 = l

neg
limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: ¬0 = l

neg
limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: ¬0 = l

neg.h
False
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: ¬0 = l

neg
limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l

neg.h
False
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: ¬0 = l

neg
limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l

neg.h
False
-- We use the *KL_bounded_imp_bounded_KSD* to extract 0 < γ < KSD(μ_t | π)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
KSD_bounded: β, ( (t : ℝ0), β < ENNReal.ofReal (KSD (μ_t t) π)) 0 < β β

neg.h
False
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: ¬0 = l

neg
limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ

neg.h.intro.intro.intro
False
-- We prove that γ/2KL(μ_0 || π) < KSD(μ_t | π) / 2KL(μ_t | π) (we plan on using it with the Stein log-Sobolev inequality).
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)

neg.h.intro.intro.intro
False
-- γ/2KL(μ_0 || π) ≠ 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)

neg.h.intro.intro.intro
False

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)

γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)

γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)

γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)

γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)

2 * KL (μ_t 0) (dμ_t 0) dπ
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)

2 * KL (μ_t 0) (dμ_t 0) dπ
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)

2 * KL (μ_t 0) (dμ_t 0) dπ
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)

2 * KL (μ_t 0) (dμ_t 0) dπ
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)

KL (μ_t 0) (dμ_t 0) dπ + KL (μ_t 0) (dμ_t 0) dπ
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)

KL (μ_t 0) (dμ_t 0) dπ + KL (μ_t 0) (dμ_t 0) dπ
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)

2 * KL (μ_t 0) (dμ_t 0) dπ

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)

γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
two_KL_finite: 2 * KL (μ_t 0) (dμ_t 0) dπ
inv_two_KL_neq: (2 * KL (μ_t 0) (dμ_t 0) dπ)⁻¹ 0

γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)

γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
two_KL_finite: 2 * KL (μ_t 0) (dμ_t 0) dπ
inv_two_KL_neq: (2 * KL (μ_t 0) (dμ_t 0) dπ)⁻¹ 0
γ_neq: γ 0

γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)

γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
two_KL_finite: 2 * KL (μ_t 0) (dμ_t 0) dπ
inv_two_KL_neq: (2 * KL (μ_t 0) (dμ_t 0) dπ)⁻¹ 0
γ_neq: γ 0

γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
two_KL_finite: 2 * KL (μ_t 0) (dμ_t 0) dπ
inv_two_KL_neq: (2 * KL (μ_t 0) (dμ_t 0) dπ)⁻¹ 0
γ_neq: γ 0

(2 * KL (μ_t 0) (dμ_t 0) dπ)⁻¹ * γ 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
two_KL_finite: 2 * KL (μ_t 0) (dμ_t 0) dπ
inv_two_KL_neq: (2 * KL (μ_t 0) (dμ_t 0) dπ)⁻¹ 0
γ_neq: γ 0

(2 * KL (μ_t 0) (dμ_t 0) dπ)⁻¹ * γ 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)

γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0

Goals accomplished! 🐙

Goals accomplished! 🐙
-- γ/2KL(μ_0 || π) ≠ ∞
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0

neg.h.intro.intro.intro
False

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0

γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0

γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0

γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0

γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0

(2 * KL (μ_t 0) (dμ_t 0) dπ)⁻¹ * γ
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0

(2 * KL (μ_t 0) (dμ_t 0) dπ)⁻¹ * γ
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0

γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0

(2 * KL (μ_t 0) (dμ_t 0) dπ)⁻¹ * γ

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0

(2 * KL (μ_t 0) (dμ_t 0) dπ)⁻¹
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0

(2 * KL (μ_t 0) (dμ_t 0) dπ)⁻¹
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0

(2 * KL (μ_t 0) (dμ_t 0) dπ)⁻¹
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0

(2 * KL (μ_t 0) (dμ_t 0) dπ)⁻¹

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0

2 * KL (μ_t 0) (dμ_t 0) dπ 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0

2 * KL (μ_t 0) (dμ_t 0) dπ 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0

2 * KL (μ_t 0) (dμ_t 0) dπ 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0

2 * KL (μ_t 0) (dμ_t 0) dπ 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0

KL (μ_t 0) (dμ_t 0) dπ + KL (μ_t 0) (dμ_t 0) dπ 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0

KL (μ_t 0) (dμ_t 0) dπ + KL (μ_t 0) (dμ_t 0) dπ 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0

2 * KL (μ_t 0) (dμ_t 0) dπ 0
[Meta.Tactic.simp.rewrite] @ne_eq:1000, KL (μ_t 0) (dμ_t 0) dπ + KL (μ_t 0) (dμ_t 0) dπ 0 ==> ¬KL (μ_t 0) (dμ_t 0) dπ + KL (μ_t 0) (dμ_t 0) dπ = 0 [Meta.Tactic.simp.rewrite] @add_eq_zero:1000, KL (μ_t 0) (dμ_t 0) dπ + KL (μ_t 0) (dμ_t 0) dπ = 0 ==> KL (μ_t 0) (dμ_t 0) dπ = 0 KL (μ_t 0) (dμ_t 0) dπ = 0 [Meta.Tactic.simp.rewrite] and_self:1000, KL (μ_t 0) (dμ_t 0) dπ = 0 KL (μ_t 0) (dμ_t 0) dπ = 0 ==> KL (μ_t 0) (dμ_t 0) dπ = 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0

¬KL (μ_t 0) (dμ_t 0) dπ = 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0

2 * KL (μ_t 0) (dμ_t 0) dπ 0

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0

(2 * KL (μ_t 0) (dμ_t 0) dπ)⁻¹

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0

γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)

Goals accomplished! 🐙

Goals accomplished! 🐙
-- Using the Stein log-Sobolev and the gamma_star inequality, we are able to show that, ∀t, KL(μ_t || π) < 1/(2*γ/2KL(μ_0 || π)) * KSD(μ_t | π).
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)

neg.h.intro.intro.intro
False

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)

(t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)

(t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)

(t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0

KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)

(t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)

KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)

(t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)

KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)

(t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)

KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)

(a b : ℝ0), a 0 a < b 1 / (2 * b) < 1 / (2 * a)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)

(a b : ℝ0), a 0 a < b 1 / (2 * b) < 1 / (2 * a)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)

(a b : ℝ0), a 0 a < b 1 / (2 * b) < 1 / (2 * a)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
a, b: 0
_ha: a 0
h: a < b

1 / (2 * b) < 1 / (2 * a)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)

(a b : ℝ0), a 0 a < b 1 / (2 * b) < 1 / (2 * a)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
a, b: 0
_ha: a 0
h: a < b

1 / (2 * b) < 1 / (2 * a)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
a, b: 0
_ha: a 0
h: a < b

1 / (2 * b) < (2 * a)⁻¹
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
a, b: 0
_ha: a 0
h: a < b

1 / (2 * b) < 1 / (2 * a)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
a, b: 0
_ha: a 0
h: a < b

(2 * b)⁻¹ < (2 * a)⁻¹
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
a, b: 0
_ha: a 0
h: a < b

(2 * b)⁻¹ < (2 * a)⁻¹
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)

(a b : ℝ0), a 0 a < b 1 / (2 * b) < 1 / (2 * a)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
a, b: 0
_ha: a 0
h: a < b

(2 * b)⁻¹ < (2 * a)⁻¹

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
a, b: 0
_ha: a 0
h: a < b

2 * a < 2 * b
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
a, b: 0
_ha: a 0
h: a < b

2 * a < 2 * b
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
a, b: 0
_ha: a 0
h: a < b

2 * a < 2 * b
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
a, b: 0
_ha: a 0
h: a < b

2 * a < 2 * b
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
a, b: 0
_ha: a 0
h: a < b

2 * a < 2 * b
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
a, b: 0
_ha: a 0
h: a < b

a + a < b + b
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
a, b: 0
_ha: a 0
h: a < b

a + a < b + b
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
a, b: 0
_ha: a 0
h: a < b

2 * a < 2 * b

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)

(a b : ℝ0), a 0 a < b 1 / (2 * b) < 1 / (2 * a)

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)

(t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
le_quotient: 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)))

KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)

(t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
le_quotient: 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)))

KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
le_quotient: 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)))

(a b c : ℝ0), c 0 c a < b a * c < b * c
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
le_quotient: 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)))

(a b c : ℝ0), c 0 c a < b a * c < b * c
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
le_quotient: 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)))

(a b c : ℝ0), c 0 c a < b a * c < b * c
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
le_quotient: 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)))
a, b, c: 0
hc_nn: c 0
hc_finite: c
ha: a < b

a * c < b * c
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
le_quotient: 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)))

(a b c : ℝ0), c 0 c a < b a * c < b * c

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)

(t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
le_quotient: 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)))
le_prod: (a b c : ℝ0), c 0 c a < b a * c < b * c

KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
le_quotient: 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)))
le_prod: (a b c : ℝ0), c 0 c a < b a * c < b * c

ENNReal.ofReal (KSD (μ_t t) π) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
le_quotient: 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)))
le_prod: (a b c : ℝ0), c 0 c a < b a * c < b * c

ENNReal.ofReal (KSD (μ_t t) π) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
le_quotient: 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)))
le_prod: (a b c : ℝ0), c 0 c a < b a * c < b * c

ENNReal.ofReal (KSD (μ_t t) π) 0

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)

(t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
le_quotient: 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)))
le_prod: (a b c : ℝ0), c 0 c a < b a * c < b * c
enn_KSD_neq: ENNReal.ofReal (KSD (μ_t t) π) 0

KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
le_quotient: 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)))
le_prod: (a b c : ℝ0), c 0 c a < b a * c < b * c
enn_KSD_neq: ENNReal.ofReal (KSD (μ_t t) π) 0

ENNReal.ofReal (KSD (μ_t t) π)
[Meta.Tactic.simp.rewrite] @ne_eq:1000, ENNReal.ofReal (KSD (μ_t t) π) ==> ¬ENNReal.ofReal (KSD (μ_t t) π) = [Meta.Tactic.simp.rewrite] @ENNReal.ofReal_ne_top:1000, ENNReal.ofReal (KSD (μ_t t) π) = ==> False [Meta.Tactic.simp.rewrite] not_false_eq_true:1000, ¬False ==> True

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)

(t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
t: 0
log_sobolev: KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
gamma_star: γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
le_quotient: 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)))
enn_KSD_neq: ENNReal.ofReal (KSD (μ_t t) π) 0
enn_KSD_finite: ENNReal.ofReal (KSD (μ_t t) π)
le_prod: 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) * ENNReal.ofReal (KSD (μ_t t) π) < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)

(t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

Goals accomplished! 🐙

Goals accomplished! 🐙
-- We use the previous inequality and the Gronwall's lemma to show that ∀t, KL(μ_t || π) ≤ KL (μ_0 || π) * exp(-2t γ/2KL(μ_0 || π)).
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

neg.h.intro.intro.intro
False

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

t > 0, KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

t > 0, KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)
-- We show that, under some non-zero and finite conditions, a ≤ (1 / (2 * c)) * b → - (b : ℝ) ≤ -2 * (c : ℝ) * (a : ℝ)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

t > 0, KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

(a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

(a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

(a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a 1 / (2 * c) * b

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

(a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a 1 / (2 * c) * b

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a 1 / (2 * c) * b

1 / (2 * c) * b = (2 * c)⁻¹ * b
[Meta.Tactic.simp.rewrite] @one_div:1000, 1 / (2 * c) ==> (2 * c)⁻¹ [Meta.Tactic.simp.rewrite] @eq_self:1000, (2 * c)⁻¹ * b = (2 * c)⁻¹ * b ==> True

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

(a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a 1 / (2 * c) * b
t: 1 / (2 * c) * b = (2 * c)⁻¹ * b

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t: 1 / (2 * c) * b = (2 * c)⁻¹ * b

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t: 1 / (2 * c) * b = (2 * c)⁻¹ * b

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t: 1 / (2 * c) * b = (2 * c)⁻¹ * b

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

(a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t: 1 / (2 * c) * b = (2 * c)⁻¹ * b

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t: 1 / (2 * c) * b = (2 * c)⁻¹ * b

2

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

(a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c

2 0
[Meta.Tactic.simp.rewrite] @ne_eq:1000, 2 0 ==> ¬2 = 0 [Meta.Tactic.simp.rewrite] @OfNat.ofNat_ne_zero:1000, 2 = 0 ==> False [Meta.Tactic.simp.rewrite] not_false_eq_true:1000, ¬False ==> True

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

(a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0

a * (2 * c) (2 * c)⁻¹ * b * (2 * c)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0

a * (2 * c) (2 * c)⁻¹ * b * (2 * c)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0

a * (2 * c) (2 * c)⁻¹ * b * (2 * c)

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

(a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt: a * (2 * c) (2 * c)⁻¹ * b * (2 * c)

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt: a * (2 * c) (2 * c)⁻¹ * b * (2 * c)

(2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

(a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt: a * (2 * c) (2 * c)⁻¹ * b * (2 * c)
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt: a * (2 * c) (2 * c)⁻¹ * b * (2 * c)
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))

(2 * c)⁻¹ * (2 * c) = 1

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

(a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt: a * (2 * c) (2 * c)⁻¹ * b * (2 * c)
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t: (2 * c)⁻¹ * (2 * c) = 1

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt: a * (2 * c) b * ((2 * c)⁻¹ * (2 * c))
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t: (2 * c)⁻¹ * (2 * c) = 1

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt: a * (2 * c) (2 * c)⁻¹ * b * (2 * c)
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t: (2 * c)⁻¹ * (2 * c) = 1

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt: a * (2 * c) b * 1
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t: (2 * c)⁻¹ * (2 * c) = 1

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt: a * (2 * c) (2 * c)⁻¹ * b * (2 * c)
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t: (2 * c)⁻¹ * (2 * c) = 1

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t: (2 * c)⁻¹ * (2 * c) = 1

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t: (2 * c)⁻¹ * (2 * c) = 1

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t: (2 * c)⁻¹ * (2 * c) = 1

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

(a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t: (2 * c)⁻¹ * (2 * c) = 1

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t: (2 * c)⁻¹ * (2 * c) = 1

ENNReal.toReal (a * (2 * c)) ENNReal.toReal b
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t: (2 * c)⁻¹ * (2 * c) = 1

ENNReal.toReal (a * (2 * c)) ENNReal.toReal b
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t: (2 * c)⁻¹ * (2 * c) = 1

ENNReal.toReal (a * (2 * c)) ENNReal.toReal b

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

(a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝¹: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t✝: (2 * c)⁻¹ * (2 * c) = 1
t: ENNReal.toReal (a * (2 * c)) ENNReal.toReal b

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝¹: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t✝: (2 * c)⁻¹ * (2 * c) = 1
t: ENNReal.toReal (a * (2 * c)) ENNReal.toReal b

ENNReal.toReal (a * (2 * c)) = ENNReal.toReal a * ENNReal.toReal (2 * c)
[Meta.Tactic.simp.rewrite] @ENNReal.toReal_mul:1000, ENNReal.toReal (a * (2 * c)) ==> ENNReal.toReal a * ENNReal.toReal (2 * c) [Meta.Tactic.simp.rewrite] @ENNReal.toReal_mul:1000, ENNReal.toReal (2 * c) ==> ENNReal.toReal 2 * ENNReal.toReal c [Meta.Tactic.simp.rewrite] ENNReal.toReal_ofNat:1000, ENNReal.toReal 2 ==> 2 [Meta.Tactic.simp.rewrite] @eq_self:1000, ENNReal.toReal a * (2 * ENNReal.toReal c) = ENNReal.toReal a * (2 * ENNReal.toReal c) ==> True

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

(a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝¹: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt✝: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t✝: (2 * c)⁻¹ * (2 * c) = 1
t: ENNReal.toReal (a * (2 * c)) ENNReal.toReal b
tt: ENNReal.toReal (a * (2 * c)) = ENNReal.toReal a * ENNReal.toReal (2 * c)

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝¹: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt✝: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t✝: (2 * c)⁻¹ * (2 * c) = 1
t: ENNReal.toReal a * ENNReal.toReal (2 * c) ENNReal.toReal b
tt: ENNReal.toReal (a * (2 * c)) = ENNReal.toReal a * ENNReal.toReal (2 * c)

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝¹: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt✝: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t✝: (2 * c)⁻¹ * (2 * c) = 1
t: ENNReal.toReal a * ENNReal.toReal (2 * c) ENNReal.toReal b
tt: ENNReal.toReal (a * (2 * c)) = ENNReal.toReal a * ENNReal.toReal (2 * c)

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝¹: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt✝: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t✝: (2 * c)⁻¹ * (2 * c) = 1
t: ENNReal.toReal a * ENNReal.toReal (2 * c) ENNReal.toReal b
tt: ENNReal.toReal (a * (2 * c)) = ENNReal.toReal a * ENNReal.toReal (2 * c)

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

(a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝¹: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt✝: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t✝: (2 * c)⁻¹ * (2 * c) = 1
t: ENNReal.toReal a * ENNReal.toReal (2 * c) ENNReal.toReal b
tt: ENNReal.toReal (a * (2 * c)) = ENNReal.toReal a * ENNReal.toReal (2 * c)

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝¹: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt✝: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t✝: (2 * c)⁻¹ * (2 * c) = 1
t: ENNReal.toReal a * ENNReal.toReal (2 * c) ENNReal.toReal b
tt: ENNReal.toReal (a * (2 * c)) = ENNReal.toReal a * ENNReal.toReal (2 * c)

ENNReal.toReal (2 * c) = ENNReal.toReal 2 * ENNReal.toReal c
[Meta.Tactic.simp.rewrite] @ENNReal.toReal_mul:1000, ENNReal.toReal (2 * c) ==> ENNReal.toReal 2 * ENNReal.toReal c [Meta.Tactic.simp.rewrite] ENNReal.toReal_ofNat:1000, ENNReal.toReal 2 ==> 2 [Meta.Tactic.simp.rewrite] @eq_self:1000, 2 * ENNReal.toReal c = 2 * ENNReal.toReal c ==> True

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

(a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝¹: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt✝¹: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t✝: (2 * c)⁻¹ * (2 * c) = 1
t: ENNReal.toReal a * ENNReal.toReal (2 * c) ENNReal.toReal b
tt✝: ENNReal.toReal (a * (2 * c)) = ENNReal.toReal a * ENNReal.toReal (2 * c)
tt: ENNReal.toReal (2 * c) = ENNReal.toReal 2 * ENNReal.toReal c

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝¹: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt✝¹: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t✝: (2 * c)⁻¹ * (2 * c) = 1
t: ENNReal.toReal a * (ENNReal.toReal 2 * ENNReal.toReal c) ENNReal.toReal b
tt✝: ENNReal.toReal (a * (2 * c)) = ENNReal.toReal a * ENNReal.toReal (2 * c)
tt: ENNReal.toReal (2 * c) = ENNReal.toReal 2 * ENNReal.toReal c

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝¹: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt✝¹: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t✝: (2 * c)⁻¹ * (2 * c) = 1
t: ENNReal.toReal a * (ENNReal.toReal 2 * ENNReal.toReal c) ENNReal.toReal b
tt✝: ENNReal.toReal (a * (2 * c)) = ENNReal.toReal a * ENNReal.toReal (2 * c)
tt: ENNReal.toReal (2 * c) = ENNReal.toReal 2 * ENNReal.toReal c

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝¹: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt✝¹: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t✝: (2 * c)⁻¹ * (2 * c) = 1
t: ENNReal.toReal a * (ENNReal.toReal 2 * ENNReal.toReal c) ENNReal.toReal b
tt✝: ENNReal.toReal (a * (2 * c)) = ENNReal.toReal a * ENNReal.toReal (2 * c)
tt: ENNReal.toReal (2 * c) = ENNReal.toReal 2 * ENNReal.toReal c

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

(a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝¹: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt✝¹: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t✝: (2 * c)⁻¹ * (2 * c) = 1
t: ENNReal.toReal a * (ENNReal.toReal 2 * ENNReal.toReal c) ENNReal.toReal b
tt✝: ENNReal.toReal (a * (2 * c)) = ENNReal.toReal a * ENNReal.toReal (2 * c)
tt: ENNReal.toReal (2 * c) = ENNReal.toReal 2 * ENNReal.toReal c

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝¹: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt✝¹: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t✝: (2 * c)⁻¹ * (2 * c) = 1
t: ENNReal.toReal a * (ENNReal.toReal 2 * ENNReal.toReal c) ENNReal.toReal b
tt✝: ENNReal.toReal (a * (2 * c)) = ENNReal.toReal a * ENNReal.toReal (2 * c)
tt: ENNReal.toReal (2 * c) = ENNReal.toReal 2 * ENNReal.toReal c

ENNReal.toReal a * (ENNReal.toReal 2 * ENNReal.toReal c) = ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

(a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝¹: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt✝²: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t✝: (2 * c)⁻¹ * (2 * c) = 1
t: ENNReal.toReal a * (ENNReal.toReal 2 * ENNReal.toReal c) ENNReal.toReal b
tt✝¹: ENNReal.toReal (a * (2 * c)) = ENNReal.toReal a * ENNReal.toReal (2 * c)
tt✝: ENNReal.toReal (2 * c) = ENNReal.toReal 2 * ENNReal.toReal c
tt: ENNReal.toReal a * (ENNReal.toReal 2 * ENNReal.toReal c) = ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝¹: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt✝²: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t✝: (2 * c)⁻¹ * (2 * c) = 1
t: ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c ENNReal.toReal b
tt✝¹: ENNReal.toReal (a * (2 * c)) = ENNReal.toReal a * ENNReal.toReal (2 * c)
tt✝: ENNReal.toReal (2 * c) = ENNReal.toReal 2 * ENNReal.toReal c
tt: ENNReal.toReal a * (ENNReal.toReal 2 * ENNReal.toReal c) = ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝¹: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt✝²: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t✝: (2 * c)⁻¹ * (2 * c) = 1
t: ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c ENNReal.toReal b
tt✝¹: ENNReal.toReal (a * (2 * c)) = ENNReal.toReal a * ENNReal.toReal (2 * c)
tt✝: ENNReal.toReal (2 * c) = ENNReal.toReal 2 * ENNReal.toReal c
tt: ENNReal.toReal a * (ENNReal.toReal 2 * ENNReal.toReal c) = ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝¹: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt✝²: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t✝: (2 * c)⁻¹ * (2 * c) = 1
t: ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c ENNReal.toReal b
tt✝¹: ENNReal.toReal (a * (2 * c)) = ENNReal.toReal a * ENNReal.toReal (2 * c)
tt✝: ENNReal.toReal (2 * c) = ENNReal.toReal 2 * ENNReal.toReal c
tt: ENNReal.toReal a * (ENNReal.toReal 2 * ENNReal.toReal c) = ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

(a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝¹: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt✝³: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t✝: (2 * c)⁻¹ * (2 * c) = 1
t: ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c ENNReal.toReal b
tt✝²: ENNReal.toReal (a * (2 * c)) = ENNReal.toReal a * ENNReal.toReal (2 * c)
tt✝¹: ENNReal.toReal (2 * c) = ENNReal.toReal 2 * ENNReal.toReal c
tt✝: ENNReal.toReal a * (ENNReal.toReal 2 * ENNReal.toReal c) = ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c
tt: -ENNReal.toReal b -(ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c)

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

(a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝¹: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt✝³: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t✝: (2 * c)⁻¹ * (2 * c) = 1
t: ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c ENNReal.toReal b
tt✝²: ENNReal.toReal (a * (2 * c)) = ENNReal.toReal a * ENNReal.toReal (2 * c)
tt✝¹: ENNReal.toReal (2 * c) = ENNReal.toReal 2 * ENNReal.toReal c
tt✝: ENNReal.toReal a * (ENNReal.toReal 2 * ENNReal.toReal c) = ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c
tt: -ENNReal.toReal b -(ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c)

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝¹: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt✝³: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t✝: (2 * c)⁻¹ * (2 * c) = 1
t: ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c ENNReal.toReal b
tt✝²: ENNReal.toReal (a * (2 * c)) = ENNReal.toReal a * ENNReal.toReal (2 * c)
tt✝¹: ENNReal.toReal (2 * c) = ENNReal.toReal 2 * ENNReal.toReal c
tt✝: ENNReal.toReal a * (ENNReal.toReal 2 * ENNReal.toReal c) = ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c
tt: -ENNReal.toReal b -(ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c)

-(ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c) = -ENNReal.toReal 2 * ENNReal.toReal c * ENNReal.toReal a

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

(a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝²: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt✝³: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t✝¹: (2 * c)⁻¹ * (2 * c) = 1
t✝: ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c ENNReal.toReal b
tt✝²: ENNReal.toReal (a * (2 * c)) = ENNReal.toReal a * ENNReal.toReal (2 * c)
tt✝¹: ENNReal.toReal (2 * c) = ENNReal.toReal 2 * ENNReal.toReal c
tt✝: ENNReal.toReal a * (ENNReal.toReal 2 * ENNReal.toReal c) = ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c
tt: -ENNReal.toReal b -(ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c)
t: -(ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c) = -ENNReal.toReal 2 * ENNReal.toReal c * ENNReal.toReal a

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝²: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt✝³: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t✝¹: (2 * c)⁻¹ * (2 * c) = 1
t✝: ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c ENNReal.toReal b
tt✝²: ENNReal.toReal (a * (2 * c)) = ENNReal.toReal a * ENNReal.toReal (2 * c)
tt✝¹: ENNReal.toReal (2 * c) = ENNReal.toReal 2 * ENNReal.toReal c
tt✝: ENNReal.toReal a * (ENNReal.toReal 2 * ENNReal.toReal c) = ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c
tt: -ENNReal.toReal b -ENNReal.toReal 2 * ENNReal.toReal c * ENNReal.toReal a
t: -(ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c) = -ENNReal.toReal 2 * ENNReal.toReal c * ENNReal.toReal a

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝²: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt✝³: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t✝¹: (2 * c)⁻¹ * (2 * c) = 1
t✝: ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c ENNReal.toReal b
tt✝²: ENNReal.toReal (a * (2 * c)) = ENNReal.toReal a * ENNReal.toReal (2 * c)
tt✝¹: ENNReal.toReal (2 * c) = ENNReal.toReal 2 * ENNReal.toReal c
tt✝: ENNReal.toReal a * (ENNReal.toReal 2 * ENNReal.toReal c) = ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c
tt: -ENNReal.toReal b -ENNReal.toReal 2 * ENNReal.toReal c * ENNReal.toReal a
t: -(ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c) = -ENNReal.toReal 2 * ENNReal.toReal c * ENNReal.toReal a

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
a, b, c: 0
htb: b
h0c: c 0
htc: c
h: a (2 * c)⁻¹ * b
t✝²: 1 / (2 * c) * b = (2 * c)⁻¹ * b
finite: 2 * c
n_zero: 2 * c 0
tt✝³: a * (2 * c) b
ttt: (2 * c)⁻¹ * b * (2 * c) = b * ((2 * c)⁻¹ * (2 * c))
t✝¹: (2 * c)⁻¹ * (2 * c) = 1
t✝: ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c ENNReal.toReal b
tt✝²: ENNReal.toReal (a * (2 * c)) = ENNReal.toReal a * ENNReal.toReal (2 * c)
tt✝¹: ENNReal.toReal (2 * c) = ENNReal.toReal 2 * ENNReal.toReal c
tt✝: ENNReal.toReal a * (ENNReal.toReal 2 * ENNReal.toReal c) = ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c
tt: -ENNReal.toReal b -ENNReal.toReal 2 * ENNReal.toReal c * ENNReal.toReal a
t: -(ENNReal.toReal a * ENNReal.toReal 2 * ENNReal.toReal c) = -ENNReal.toReal 2 * ENNReal.toReal c * ENNReal.toReal a

-ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

(a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

t > 0, KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
calculation: (a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
t: 0
pos_t: t > 0

KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)
-- This calculation allows us to transform the inequality KL(μ_t || π) < 1/(2*γ/2KL(μ_0 || π)) * KSD(μ_t | π) into -KSD(μ_t | π) ≤ -2 * γ/2KL(μ_0 || π) * KL(μ_t || π)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
calculation: (a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
t: 0
pos_t: t > 0

KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
calculation: (a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
t: 0
pos_t: t > 0

ENNReal.ofReal (KSD (μ_t t) π)
[Meta.Tactic.simp.rewrite] @ne_eq:1000, ENNReal.ofReal (KSD (μ_t t) π) ==> ¬ENNReal.ofReal (KSD (μ_t t) π) = [Meta.Tactic.simp.rewrite] @ENNReal.ofReal_ne_top:1000, ENNReal.ofReal (KSD (μ_t t) π) = ==> False [Meta.Tactic.simp.rewrite] not_false_eq_true:1000, ¬False ==> True

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
calculation: (a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
t: 0
pos_t: t > 0
compute_ineq: -ENNReal.toReal (ENNReal.ofReal (KSD (μ_t t) π)) -2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)

KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)

t > 0, KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
calculation: (a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
t: 0
pos_t: t > 0
compute_ineq: -ENNReal.toReal (ENNReal.ofReal (KSD (μ_t t) π)) -2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)

KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
calculation: (a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
t: 0
pos_t: t > 0
compute_ineq: -KSD (μ_t t) π -2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)

KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
calculation: (a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
t: 0
pos_t: t > 0
compute_ineq: -KSD (μ_t t) π -2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)

KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
calculation: (a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
t: 0
pos_t: t > 0
compute_ineq: -KSD (μ_t t) π -2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)

KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)
-- As d_KL_t t ≤ -KSD (μ_t t | π) and -KSD (μ_t t | π) ≤ -2 * γ/2KL(μ_0 || π) * KL(μ_t t || π), then d_KL_t t ≤ -2 * γ/2KL(μ_0 || π) * KL(μ_t t || π).
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
calculation: (a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
t: 0
pos_t: t > 0
compute_ineq: -KSD (μ_t t) π -2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)
dkl_ineq: d_KL_t t -2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)

KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)
-- We finally can use the Gronwall's lemma with ψ := t ↦ -2 * γ/2KL(μ_0 || π).
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
calculation: (a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
t: 0
pos_t: t > 0
compute_ineq: -KSD (μ_t t) π -2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)
dkl_ineq: d_KL_t t -2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)

KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)
Warning: unused variable `t` [linter.unusedVariables]
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
calculation: (a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
t: 0
pos_t: t > 0
compute_ineq: -KSD (μ_t t) π -2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)
dkl_ineq: d_KL_t t -2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)

KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
calculation: (a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
t: 0
pos_t: t > 0
compute_ineq: -KSD (μ_t t) π -2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)
dkl_ineq: d_KL_t t -2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)
gronwall: KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), -2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)))

KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)
-- We rewrite ∫ s ∈ [0, t], -2 * γ/2KL(μ_0 || π) dt as -2t γ/2KL(μ_0 || π).
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
calculation: (a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
t: 0
pos_t: t > 0
compute_ineq: -KSD (μ_t t) π -2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)
dkl_ineq: d_KL_t t -2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)
gronwall: KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), -2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)))

KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
calculation: (a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
t: 0
pos_t: t > 0
compute_ineq: -KSD (μ_t t) π -2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)
dkl_ineq: d_KL_t t -2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)
gronwall: KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)

KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
calculation: (a b c : ℝ0), b c 0 c a 1 / (2 * c) * b -ENNReal.toReal b -2 * ENNReal.toReal c * ENNReal.toReal a
t: 0
pos_t: t > 0
compute_ineq: -KSD (μ_t t) π -2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)
dkl_ineq: d_KL_t t -2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)
gronwall: KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)

KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)

Goals accomplished! 🐙

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: ¬0 = l

neg
limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
bound_gronwall: t > 0, KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)

neg.h.intro.intro.intro
False

Goals accomplished! 🐙
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
bound_gronwall: t > 0, KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)

-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) < 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
bound_gronwall: t > 0, KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)

-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) < 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
bound_gronwall: t > 0, KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)

-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) < 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
bound_gronwall: t > 0, KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)
gamma_star_nn: 0 < γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)

-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) < 0
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
bound_gronwall: t > 0, KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)

-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) < 0
[Meta.Tactic.simp.rewrite] @neg_mul:1000, -2 * ENNReal.toReal (γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)) ==> -(2 * ENNReal.toReal (γ / (2 * KL (μ_t 0) (dμ_t 0) dπ))) [Meta.Tactic.simp.rewrite] @Left.neg_neg_iff:1000, -(2 * ENNReal.toReal (γ / (2 * KL (μ_t 0) (dμ_t 0) dπ))) < 0 ==> 0 < 2 * ENNReal.toReal (γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)) [Meta.Tactic.simp.rewrite] @gt_iff_lt:1000, 0 < 2 ==> 0 < 2 [Meta.Tactic.simp.rewrite] @zero_lt_two:1000, 0 < 2 ==> True [Meta.Tactic.simp.rewrite] @zero_lt_mul_left:1000, 0 < 2 * ENNReal.toReal (γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)) ==> 0 < ENNReal.toReal (γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)) [Meta.Tactic.simp.rewrite] @gt_iff_lt:1000, 0 < ENNReal.toReal (γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)) ==> 0 < ENNReal.toReal (γ / (2 * KL (μ_t 0) (dμ_t 0) dπ))
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
bound_gronwall: t > 0, KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)
gamma_star_nn: 0 < γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)

0 < ENNReal.toReal (γ / (2 * KL (μ_t 0) (dμ_t 0) dπ))
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
bound_gronwall: t > 0, KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)

-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) < 0

Goals accomplished! 🐙

Goals accomplished! 🐙
-- As -2 γ/2KL(μ_0 || π) < 0, lim_(t → ∞) exp(-2 γ/2KL(μ_0 || π) t) = 0.
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
bound_gronwall: t > 0, KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)
minus_gamma_star_neg: -2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) < 0
exp_limit: limit (fun t => KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)) 0

neg.h.intro.intro.intro
False
-- Using the squeeze theorem and the previous result, we prove that lim_(t → ∞) KL(μ_0 || π) t) = 0.
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
bound_gronwall: t > 0, KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)
minus_gamma_star_neg: -2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) < 0
exp_limit: limit (fun t => KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)) 0
contradiction_limit: limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0

neg.h.intro.intro.intro
False
-- The last result implies that l = 0 which is a contradiction as we supposed l ≠ 0.
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: 0 l
lim_nn: 0 < l
γ: 0
KSD_bounded: (t : ℝ0), γ < ENNReal.ofReal (KSD (μ_t t) π)
γ_nn: 0 < γ
γ_finite: γ
gamma_star: (t : ℝ0), γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0) < (fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
gamma_star_neq: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ) 0
gamma_star_finite: γ / (2 * KL (μ_t 0) (dμ_t 0) dπ)
bounded_log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ < 1 / (2 */ (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0))) * ENNReal.ofReal (KSD (μ_t t) π)
bound_gronwall: t > 0, KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)
minus_gamma_star_neg: -2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) < 0
exp_limit: limit (fun t => KL (μ_t 0) (dμ_t 0) dπ * exp (-2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0)) * t)) 0
contradiction_limit: limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
lim_eq_zero: l = 0

neg.h.intro.intro.intro
False
d:
inst✝¹³: MeasurableSpace (Vector ℝ d)
inst✝¹²: MeasureSpace (Vector ℝ d)
inst✝¹¹: MeasureSpace ℝ
μ, π, ν: Measure (Vector ℝ d)
dμ, dπ: Vector ℝ d 0
_h1: absolutely_continuous μ π
_h2: absolutely_continuous π ν
: is_density μ ν dμ
: is_density π ν dπ
mdμ: Measurable dμ
mdπ: Measurable dπ
hdμ: (x : Vector ℝ d), dμ x 0 dμ x
hdπ: (x : Vector ℝ d), dπ x 0 dπ x
inst✝¹⁰: IsProbabilityMeasure μ
inst✝⁹: IsProbabilityMeasure π
h_m_set: (s : Set (Vector ℝ d)), MeasurableSet s
H₀: Set (Vector ℝ d ℝ)
inst✝⁸: NormedAddCommGroup (Vector ℝ d ℝ)
inst✝⁷: InnerProductSpace ℝ (Vector ℝ d ℝ)
k: Vector ℝ d Vector ℝ d
h_k: ( (x : Vector ℝ d), k x H₀) (x : Vector ℝ d), (fun y => k y x) H₀
h_kernel: is_kernel H₀ k
h_kernel_positive: positive_definite_kernel μ k
H: Set (ℕ Vector ℝ d ℝ)
inst✝⁶: NormedAddCommGroup (ℕ Vector ℝ d ℝ)
inst✝⁵: InnerProductSpace ℝ (ℕ Vector ℝ d ℝ)
dk: Vector ℝ d Vector ℝ d
d_log_π, ϕ: Vector ℝ d
: ϕ H
: Vector ℝ d
h_is_ϕ: is_phi μ k dk d_log_π ϕ
is_integrable_H₀: (f : Vector ℝ d ℝ), Integrable f
d_log_π_μ: Vector ℝ d
hd_log_π_μ: ( (x : Vector ℝ d) (i : ℕ), d_log_π_μ i x = 0) c, (x : Vector ℝ d), log (dμ x / dπ x) = c
dπ': Vector ℝ d
hπ': (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_log_π i x = dπ' i x
inst✝⁴: Norm (Vector ℝ d)
KSD: Measure (Vector ℝ d) Measure (Vector ℝ d)
ksd_nn: 0 KSD μ π
hstein: SteinClass ϕ dπ
hkl_eq: μ = π KL μ dμ dπ = 0
hkl_diff: μ π 0 < KL μ dμ dπ
μ_t: 0 Measure (Vector ℝ d)
dμ_t: 0 Vector ℝ d 0
hμ_t: (t : ℝ0), is_density (μ_t t) ν (dμ_t t)
h_prob: (t : ℝ0), IsProbabilityMeasure (μ_t t)
hdμ_t: (t : ℝ0) (x : Vector ℝ d), dμ_t t x 0 dμ_t t x
d_KL_t: 0
d_log_μ_t_π: 0 Vector ℝ d
hd_log_μ_t_π: (t : ℝ0), ( (x : Vector ℝ d) (i : ℕ), d_log_μ_t_π t i x = 0) c, (x : Vector ℝ d), log (dμ_t t x / dπ x) = c
hkl_eq_t: (t : ℝ0), μ_t t = π KL (μ_t t) (dμ_t t) dπ = 0
hkl_diff_t: (t : ℝ0), μ_t t π 0 < KL (μ_t t) (dμ_t t) dπ
h_kernel_positive_t: (t : ℝ0), positive_definite_kernel (μ_t t) k
is_integrable_H₀_t: (t : ℝ0) (f : Vector ℝ d ℝ), Integrable f
ksd_nn_t: (t : ℝ0), 0 KSD (μ_t t) π
inst✝³: MeasureSpace ℝ0
inst✝²: NormedAddCommGroup ℝ0
inst✝¹: NormedSpace ℝ ℝ0
inst✝: LocallyFiniteOrder ℝ0
gronwall: (ψ : ℝ0 ℝ), t > 0, d_KL_t t ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) KL (μ_t t) (dμ_t t) dπ KL (μ_t 0) (dμ_t 0) dπ * exp ( (s : ℝ0) in (Icc 0 t), ψ s)
dkl_ksd: (t : ℝ0), d_KL_t t -KSD (μ_t t) π
kl_decreasing: (t t' : ℝ0), t < t' KL (μ_t t') (dμ_t t') dπ KL (μ_t t) (dμ_t t) dπ
kl_finite: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ
log_sobolev: (t : ℝ0), KL (μ_t t) (dμ_t t) dπ 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
l: 0
lim: limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
_lim_pos: 0 l
KL_bounded: (x : ℝ0), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
hl: ¬0 = l

neg
limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0

Goals accomplished! 🐙

Goals accomplished! 🐙